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Properties

Label 4-105e2-1.1-c3e2-0-2
Degree 44
Conductor 1102511025
Sign 11
Analytic cond. 38.380538.3805
Root an. cond. 2.489012.48901
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 6·3-s + 4-s + 10·5-s + 6·6-s − 14·7-s + 9·8-s + 27·9-s + 10·10-s − 22·11-s + 6·12-s − 22·13-s − 14·14-s + 60·15-s − 47·16-s + 116·17-s + 27·18-s + 102·19-s + 10·20-s − 84·21-s − 22·22-s + 260·23-s + 54·24-s + 75·25-s − 22·26-s + 108·27-s − 14·28-s + ⋯
L(s)  = 1  + 0.353·2-s + 1.15·3-s + 1/8·4-s + 0.894·5-s + 0.408·6-s − 0.755·7-s + 0.397·8-s + 9-s + 0.316·10-s − 0.603·11-s + 0.144·12-s − 0.469·13-s − 0.267·14-s + 1.03·15-s − 0.734·16-s + 1.65·17-s + 0.353·18-s + 1.23·19-s + 0.111·20-s − 0.872·21-s − 0.213·22-s + 2.35·23-s + 0.459·24-s + 3/5·25-s − 0.165·26-s + 0.769·27-s − 0.0944·28-s + ⋯

Functional equation

Λ(s)=(11025s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(11025s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1102511025    =    32⋅52⋅723^{2} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 38.380538.3805
Root analytic conductor: 2.489012.48901
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 11025, ( :3/2,3/2), 1)(4,\ 11025,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 4.4475509524.447550952
L(12)L(\frac12) ≈\approx 4.4475509524.447550952
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3C1C_1 (1−pT)2 ( 1 - p T )^{2}
5C1C_1 (1−pT)2 ( 1 - p T )^{2}
7C1C_1 (1+pT)2 ( 1 + p T )^{2}
good2D4D_{4} 1−T−p3T3+p6T4 1 - T - p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+2pT+2718T2+2p4T3+p6T4 1 + 2 p T + 2718 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4}
13D4D_{4} 1+22T+4450T2+22p3T3+p6T4 1 + 22 T + 4450 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1−116T+9030T2−116p3T3+p6T4 1 - 116 T + 9030 T^{2} - 116 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1−102T+13134T2−102p3T3+p6T4 1 - 102 T + 13134 T^{2} - 102 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−260T+34734T2−260p3T3+p6T4 1 - 260 T + 34734 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+196T+20942T2+196p3T3+p6T4 1 + 196 T + 20942 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−150T+36542T2−150p3T3+p6T4 1 - 150 T + 36542 T^{2} - 150 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+96T+82550T2+96p3T3+p6T4 1 + 96 T + 82550 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+176T−16914T2+176p3T3+p6T4 1 + 176 T - 16914 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1+8pT+171958T2+8p4T3+p6T4 1 + 8 p T + 171958 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4}
47D4D_{4} 1−560T+248606T2−560p3T3+p6T4 1 - 560 T + 248606 T^{2} - 560 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−326T+204138T2−326p3T3+p6T4 1 - 326 T + 204138 T^{2} - 326 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+844T+474182T2+844p3T3+p6T4 1 + 844 T + 474182 T^{2} + 844 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+204T+455006T2+204p3T3+p6T4 1 + 204 T + 455006 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+104T+537670T2+104p3T3+p6T4 1 + 104 T + 537670 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1−1670T+1384382T2−1670p3T3+p6T4 1 - 1670 T + 1384382 T^{2} - 1670 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+386T+152218T2+386p3T3+p6T4 1 + 386 T + 152218 T^{2} + 386 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+888T+1007454T2+888p3T3+p6T4 1 + 888 T + 1007454 T^{2} + 888 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−928T+600710T2−928p3T3+p6T4 1 - 928 T + 600710 T^{2} - 928 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−588T+1495334T2−588p3T3+p6T4 1 - 588 T + 1495334 T^{2} - 588 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−522T+291282T2−522p3T3+p6T4 1 - 522 T + 291282 T^{2} - 522 p^{3} T^{3} + p^{6} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−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

−13.40546592608703245871921898326, −13.30346175272847329227780832686, −12.73831602913954178485104586965, −12.18250555097709804072936119577, −11.53088546756880426510015652679, −10.67678270051248069312468546548, −10.24689496600514792997960725745, −9.750487568024786448080816428638, −9.153033322238177666395213432806, −8.959543556742975218076338319296, −7.920982822273216246053602816816, −7.35197583697436006371302737442, −7.02876736555687686837865597329, −6.13911297908769492799448079281, −5.17693363736470717283113350431, −4.97877979516232912120258346709, −3.60040878612527266076080746053, −3.08354544168682833867216834671, −2.36836063550997595589839875669, −1.18419157274904927371218082605, 1.18419157274904927371218082605, 2.36836063550997595589839875669, 3.08354544168682833867216834671, 3.60040878612527266076080746053, 4.97877979516232912120258346709, 5.17693363736470717283113350431, 6.13911297908769492799448079281, 7.02876736555687686837865597329, 7.35197583697436006371302737442, 7.920982822273216246053602816816, 8.959543556742975218076338319296, 9.153033322238177666395213432806, 9.750487568024786448080816428638, 10.24689496600514792997960725745, 10.67678270051248069312468546548, 11.53088546756880426510015652679, 12.18250555097709804072936119577, 12.73831602913954178485104586965, 13.30346175272847329227780832686, 13.40546592608703245871921898326

Graph of the ZZ-function along the critical line