Properties

Label 4-48e2-1.1-c27e2-0-2
Degree 44
Conductor 23042304
Sign 11
Analytic cond. 49146.749146.7
Root an. cond. 14.889214.8892
Motivic weight 2727
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.18e6·3-s + 2.91e8·5-s − 1.21e11·7-s + 7.62e12·9-s + 2.31e14·11-s − 1.15e15·13-s + 9.29e14·15-s − 4.61e16·17-s + 2.68e17·19-s − 3.87e17·21-s − 1.29e18·23-s + 3.00e18·25-s + 1.62e19·27-s + 6.01e19·29-s − 2.03e20·31-s + 7.39e20·33-s − 3.54e19·35-s + 3.02e21·37-s − 3.68e21·39-s − 1.08e22·41-s + 1.29e21·43-s + 2.22e21·45-s + 9.93e22·47-s − 5.76e22·49-s − 1.47e23·51-s − 3.57e22·53-s + 6.75e22·55-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.106·5-s − 0.474·7-s + 9-s + 2.02·11-s − 1.05·13-s + 0.123·15-s − 1.12·17-s + 1.46·19-s − 0.547·21-s − 0.537·23-s + 0.402·25-s + 0.769·27-s + 1.08·29-s − 1.49·31-s + 2.33·33-s − 0.0506·35-s + 2.03·37-s − 1.22·39-s − 1.83·41-s + 0.115·43-s + 0.106·45-s + 2.65·47-s − 0.876·49-s − 1.30·51-s − 0.188·53-s + 0.216·55-s + ⋯

Functional equation

Λ(s)=(2304s/2ΓC(s)2L(s)=(Λ(28s)\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}
Λ(s)=(2304s/2ΓC(s+27/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+27/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 23042304    =    28322^{8} \cdot 3^{2}
Sign: 11
Analytic conductor: 49146.749146.7
Root analytic conductor: 14.889214.8892
Motivic weight: 2727
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2304, ( :27/2,27/2), 1)(4,\ 2304,\ (\ :27/2, 27/2),\ 1)

Particular Values

L(14)L(14) \approx 7.6091851097.609185109
L(12)L(\frac12) \approx 7.6091851097.609185109
L(292)L(\frac{29}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3C1C_1 (1p13T)2 ( 1 - p^{13} T )^{2}
good5D4D_{4} 1291441036T23327953942428106p3T2291441036p27T3+p54T4 1 - 291441036 T - 23327953942428106 p^{3} T^{2} - 291441036 p^{27} T^{3} + p^{54} T^{4}
7D4D_{4} 1+121646295328T+ 1 + 121646295328 T + 21 ⁣ ⁣7421\!\cdots\!74p3T2+121646295328p27T3+p54T4 p^{3} T^{2} + 121646295328 p^{27} T^{3} + p^{54} T^{4}
11D4D_{4} 121073396524216pT+ 1 - 21073396524216 p T + 27 ⁣ ⁣0627\!\cdots\!06p3T221073396524216p28T3+p54T4 p^{3} T^{2} - 21073396524216 p^{28} T^{3} + p^{54} T^{4}
13D4D_{4} 1+1155759271611764T+ 1 + 1155759271611764 T + 93 ⁣ ⁣6693\!\cdots\!66pT2+1155759271611764p27T3+p54T4 p T^{2} + 1155759271611764 p^{27} T^{3} + p^{54} T^{4}
17D4D_{4} 1+46125175777027452T+ 1 + 46125175777027452 T + 75 ⁣ ⁣6675\!\cdots\!66pT2+46125175777027452p27T3+p54T4 p T^{2} + 46125175777027452 p^{27} T^{3} + p^{54} T^{4}
19D4D_{4} 114132623934381000pT+ 1 - 14132623934381000 p T + 13 ⁣ ⁣9813\!\cdots\!98p2T214132623934381000p28T3+p54T4 p^{2} T^{2} - 14132623934381000 p^{28} T^{3} + p^{54} T^{4}
23D4D_{4} 1+56502692435040432pT+ 1 + 56502692435040432 p T + 14 ⁣ ⁣4214\!\cdots\!42p2T2+56502692435040432p28T3+p54T4 p^{2} T^{2} + 56502692435040432 p^{28} T^{3} + p^{54} T^{4}
29D4D_{4} 160178362995785587900T+ 1 - 60178362995785587900 T + 58 ⁣ ⁣1858\!\cdots\!18T260178362995785587900p27T3+p54T4 T^{2} - 60178362995785587900 p^{27} T^{3} + p^{54} T^{4}
31D4D_{4} 1+ 1 + 20 ⁣ ⁣4420\!\cdots\!44T+ T + 46 ⁣ ⁣0646\!\cdots\!06T2+ T^{2} + 20 ⁣ ⁣4420\!\cdots\!44p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
37D4D_{4} 1 1 - 30 ⁣ ⁣8830\!\cdots\!88T+ T + 58 ⁣ ⁣0258\!\cdots\!02T2 T^{2} - 30 ⁣ ⁣8830\!\cdots\!88p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
41D4D_{4} 1+ 1 + 10 ⁣ ⁣1610\!\cdots\!16T+ T + 78 ⁣ ⁣2678\!\cdots\!26T2+ T^{2} + 10 ⁣ ⁣1610\!\cdots\!16p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
43D4D_{4} 1 1 - 12 ⁣ ⁣4412\!\cdots\!44T+ T + 19 ⁣ ⁣9819\!\cdots\!98T2 T^{2} - 12 ⁣ ⁣4412\!\cdots\!44p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
47D4D_{4} 1 1 - 99 ⁣ ⁣3299\!\cdots\!32T+ T + 47 ⁣ ⁣8247\!\cdots\!82T2 T^{2} - 99 ⁣ ⁣3299\!\cdots\!32p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
53D4D_{4} 1+ 1 + 35 ⁣ ⁣2435\!\cdots\!24T+ T + 13 ⁣ ⁣1813\!\cdots\!18T2+ T^{2} + 35 ⁣ ⁣2435\!\cdots\!24p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
59D4D_{4} 1+ 1 + 29 ⁣ ⁣8029\!\cdots\!80T+ T + 12 ⁣ ⁣3812\!\cdots\!38T2+ T^{2} + 29 ⁣ ⁣8029\!\cdots\!80p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
61D4D_{4} 1 1 - 21 ⁣ ⁣2421\!\cdots\!24T+ T + 33 ⁣ ⁣8633\!\cdots\!86T2 T^{2} - 21 ⁣ ⁣2421\!\cdots\!24p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
67D4D_{4} 1+ 1 + 13 ⁣ ⁣0813\!\cdots\!08T+ T + 83 ⁣ ⁣6283\!\cdots\!62T2+ T^{2} + 13 ⁣ ⁣0813\!\cdots\!08p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
71D4D_{4} 1 1 - 43 ⁣ ⁣3643\!\cdots\!36T+ T + 70 ⁣ ⁣0670\!\cdots\!06T2 T^{2} - 43 ⁣ ⁣3643\!\cdots\!36p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
73D4D_{4} 1+ 1 + 20 ⁣ ⁣6420\!\cdots\!64T+ T + 49 ⁣ ⁣1849\!\cdots\!18T2+ T^{2} + 20 ⁣ ⁣6420\!\cdots\!64p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
79D4D_{4} 1+ 1 + 56 ⁣ ⁣0056\!\cdots\!00T+ T + 24 ⁣ ⁣1824\!\cdots\!18T2+ T^{2} + 56 ⁣ ⁣0056\!\cdots\!00p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
83D4D_{4} 1 1 - 46 ⁣ ⁣0446\!\cdots\!04T+ T + 12 ⁣ ⁣5812\!\cdots\!58T2 T^{2} - 46 ⁣ ⁣0446\!\cdots\!04p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
89D4D_{4} 1+ 1 + 19 ⁣ ⁣8019\!\cdots\!80T+ T + 45 ⁣ ⁣5845\!\cdots\!58T2+ T^{2} + 19 ⁣ ⁣8019\!\cdots\!80p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
97D4D_{4} 1 1 - 54 ⁣ ⁣6854\!\cdots\!68T+ T + 94 ⁣ ⁣8294\!\cdots\!82T2 T^{2} - 54 ⁣ ⁣6854\!\cdots\!68p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
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   L(s)=p j=14(1αj,pps)1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.94436693500532443005117157072, −10.16615095017602300676912623779, −9.690470117714450293190299982649, −9.330698331600928852650419157591, −8.865676492525466583574453318746, −8.495997507737936948496271897935, −7.48621736765211734997499098291, −7.30796439538500621827271816730, −6.70625691975758057482531836782, −6.18841066958626046258365738347, −5.50476369418244116681712293780, −4.59647152603746605198860374674, −4.28148016975867483145426968632, −3.75246637682478132245336427694, −2.96148040917870694438810242911, −2.88897734265127925217414309474, −1.81038388398900894849820668871, −1.80636949032861795819643728766, −0.865440736491405725008687668011, −0.52071941934844237508838271620, 0.52071941934844237508838271620, 0.865440736491405725008687668011, 1.80636949032861795819643728766, 1.81038388398900894849820668871, 2.88897734265127925217414309474, 2.96148040917870694438810242911, 3.75246637682478132245336427694, 4.28148016975867483145426968632, 4.59647152603746605198860374674, 5.50476369418244116681712293780, 6.18841066958626046258365738347, 6.70625691975758057482531836782, 7.30796439538500621827271816730, 7.48621736765211734997499098291, 8.495997507737936948496271897935, 8.865676492525466583574453318746, 9.330698331600928852650419157591, 9.690470117714450293190299982649, 10.16615095017602300676912623779, 10.94436693500532443005117157072

Graph of the ZZ-function along the critical line