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Properties

Label 4-312e2-1.1-c3e2-0-1
Degree 44
Conductor 9734497344
Sign 11
Analytic cond. 338.876338.876
Root an. cond. 4.290524.29052
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  + 6·3-s + 12·5-s + 44·7-s + 27·9-s + 52·11-s − 26·13-s + 72·15-s − 20·17-s − 60·19-s + 264·21-s − 8·23-s + 30·25-s + 108·27-s + 132·29-s − 140·31-s + 312·33-s + 528·35-s + 68·37-s − 156·39-s + 28·41-s + 324·45-s + 36·47-s + 938·49-s − 120·51-s + 668·53-s + 624·55-s − 360·57-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.07·5-s + 2.37·7-s + 9-s + 1.42·11-s − 0.554·13-s + 1.23·15-s − 0.285·17-s − 0.724·19-s + 2.74·21-s − 0.0725·23-s + 6/25·25-s + 0.769·27-s + 0.845·29-s − 0.811·31-s + 1.64·33-s + 2.54·35-s + 0.302·37-s − 0.640·39-s + 0.106·41-s + 1.07·45-s + 0.111·47-s + 2.73·49-s − 0.329·51-s + 1.73·53-s + 1.52·55-s − 0.836·57-s + ⋯

Functional equation

Λ(s)=(97344s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(97344s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9734497344    =    26⋅32⋅1322^{6} \cdot 3^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 338.876338.876
Root analytic conductor: 4.290524.29052
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 97344, ( :3/2,3/2), 1)(4,\ 97344,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 8.1885211398.188521139
L(12)L(\frac12) ≈\approx 8.1885211398.188521139
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)
bad2 1 1
3C1C_1 (1−pT)2 ( 1 - p T )^{2}
13C1C_1 (1+pT)2 ( 1 + p T )^{2}
good5D4D_{4} 1−12T+114T2−12p3T3+p6T4 1 - 12 T + 114 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 1−44T+998T2−44p3T3+p6T4 1 - 44 T + 998 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4}
11C2C_2 (1−26T+p3T2)2 ( 1 - 26 T + p^{3} T^{2} )^{2}
17D4D_{4} 1+20T+9238T2+20p3T3+p6T4 1 + 20 T + 9238 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+60T+10318T2+60p3T3+p6T4 1 + 60 T + 10318 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+8T−418T2+8p3T3+p6T4 1 + 8 T - 418 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1−132T+28366T2−132p3T3+p6T4 1 - 132 T + 28366 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+140T+25782T2+140p3T3+p6T4 1 + 140 T + 25782 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−68T+68750T2−68p3T3+p6T4 1 - 68 T + 68750 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−28T+129610T2−28p3T3+p6T4 1 - 28 T + 129610 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4}
43C22C_2^2 1+2914pT2+p6T4 1 + 2914 p T^{2} + p^{6} T^{4}
47D4D_{4} 1−36T+201778T2−36p3T3+p6T4 1 - 36 T + 201778 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−668T+406558T2−668p3T3+p6T4 1 - 668 T + 406558 T^{2} - 668 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+508T+392026T2+508p3T3+p6T4 1 + 508 T + 392026 T^{2} + 508 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1−340T+471854T2−340p3T3+p6T4 1 - 340 T + 471854 T^{2} - 340 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+940T+504398T2+940p3T3+p6T4 1 + 940 T + 504398 T^{2} + 940 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1−300T−57006T2−300p3T3+p6T4 1 - 300 T - 57006 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−1124T+1087686T2−1124p3T3+p6T4 1 - 1124 T + 1087686 T^{2} - 1124 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+1520T+1230686T2+1520p3T3+p6T4 1 + 1520 T + 1230686 T^{2} + 1520 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−524T+908810T2−524p3T3+p6T4 1 - 524 T + 908810 T^{2} - 524 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−1900T+2312266T2−1900p3T3+p6T4 1 - 1900 T + 2312266 T^{2} - 1900 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1+1436T+2285142T2+1436p3T3+p6T4 1 + 1436 T + 2285142 T^{2} + 1436 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

−11.42214398917128688713621417985, −11.05144205561124725340267116979, −10.31194809942975245233471190816, −10.24788617272732909805315794294, −9.264813226358060540585988578082, −9.215669671319814268434114253472, −8.606010351118545401860918348051, −8.346252110582788825116967335954, −7.54957521691544461779378250866, −7.49378800629612049326858629046, −6.53715315204666169112077639816, −6.23805535267007924477684695878, −5.23710427814114969253752179319, −5.00426301943975155280014032240, −4.15952790984489884125185039394, −3.98523586614477440481232678366, −2.76919953794383634385098601660, −2.10695978885312781704145186954, −1.71775559267031069696382642172, −1.10022464665493037666502970190, 1.10022464665493037666502970190, 1.71775559267031069696382642172, 2.10695978885312781704145186954, 2.76919953794383634385098601660, 3.98523586614477440481232678366, 4.15952790984489884125185039394, 5.00426301943975155280014032240, 5.23710427814114969253752179319, 6.23805535267007924477684695878, 6.53715315204666169112077639816, 7.49378800629612049326858629046, 7.54957521691544461779378250866, 8.346252110582788825116967335954, 8.606010351118545401860918348051, 9.215669671319814268434114253472, 9.264813226358060540585988578082, 10.24788617272732909805315794294, 10.31194809942975245233471190816, 11.05144205561124725340267116979, 11.42214398917128688713621417985

Graph of the ZZ-function along the critical line