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Properties

Label 4-3240e2-1.1-c1e2-0-15
Degree 44
Conductor 1049760010497600
Sign 11
Analytic cond. 669.336669.336
Root an. cond. 5.086405.08640
Motivic weight 11
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  + 5-s + 7-s − 2·11-s + 5·13-s + 8·17-s − 10·19-s − 2·23-s + 10·29-s + 8·31-s + 35-s − 6·37-s + 6·41-s − 4·43-s − 8·47-s + 7·49-s − 12·53-s − 2·55-s − 4·59-s + 5·61-s + 5·65-s + 7·67-s − 12·71-s − 18·73-s − 2·77-s − 3·79-s + 2·83-s + 8·85-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 0.603·11-s + 1.38·13-s + 1.94·17-s − 2.29·19-s − 0.417·23-s + 1.85·29-s + 1.43·31-s + 0.169·35-s − 0.986·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s + 49-s − 1.64·53-s − 0.269·55-s − 0.520·59-s + 0.640·61-s + 0.620·65-s + 0.855·67-s − 1.42·71-s − 2.10·73-s − 0.227·77-s − 0.337·79-s + 0.219·83-s + 0.867·85-s + ⋯

Functional equation

Λ(s)=(10497600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(10497600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1049760010497600    =    26⋅38⋅522^{6} \cdot 3^{8} \cdot 5^{2}
Sign: 11
Analytic conductor: 669.336669.336
Root analytic conductor: 5.086405.08640
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 10497600, ( :1/2,1/2), 1)(4,\ 10497600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.1104759533.110475953
L(12)L(\frac12) ≈\approx 3.1104759533.110475953
L(32)L(\frac{3}{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
3 1 1
5C2C_2 1−T+T2 1 - T + T^{2}
good7C2C_2 (1−5T+pT2)(1+4T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C22C_2^2 1+2T−7T2+2pT3+p2T4 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4}
13C2C_2 (1−7T+pT2)(1+2T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
19C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
23C22C_2^2 1+2T−19T2+2pT3+p2T4 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4}
29C22C_2^2 1−10T+71T2−10pT3+p2T4 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4}
31C22C_2^2 1−8T+33T2−8pT3+p2T4 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4}
37C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
41C22C_2^2 1−6T−5T2−6pT3+p2T4 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4}
43C22C_2^2 1+4T−27T2+4pT3+p2T4 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4}
47C22C_2^2 1+8T+17T2+8pT3+p2T4 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4}
53C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
59C22C_2^2 1+4T−43T2+4pT3+p2T4 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−5T−36T2−5pT3+p2T4 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−7T−18T2−7pT3+p2T4 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4}
71C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
73C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
79C22C_2^2 1+3T−70T2+3pT3+p2T4 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4}
83C22C_2^2 1−2T−79T2−2pT3+p2T4 1 - 2 T - 79 T^{2} - 2 p T^{3} + p^{2} T^{4}
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
97C22C_2^2 1+7T−48T2+7pT3+p2T4 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} 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

−8.712607036135152781559145359729, −8.463515552407996208283127995058, −8.104743937011490115203757494307, −7.978997333339432818276948388429, −7.32339905040966103274605126905, −7.00815450400771221949493473984, −6.40870525303038711965989394035, −6.11779289366797645606845046099, −5.97175004030071896383000131913, −5.58488565043829121858857465160, −4.75244905254776834002228683580, −4.75179072627544156992910344013, −4.31601563699528076500694039246, −3.65866775998458643963262225566, −3.22740875260221110389281653390, −2.90634465987910073595815602894, −2.24408207472354851537028761442, −1.74251376984095263425476971971, −1.24847036278373727667627163957, −0.57467501467699531115302204025, 0.57467501467699531115302204025, 1.24847036278373727667627163957, 1.74251376984095263425476971971, 2.24408207472354851537028761442, 2.90634465987910073595815602894, 3.22740875260221110389281653390, 3.65866775998458643963262225566, 4.31601563699528076500694039246, 4.75179072627544156992910344013, 4.75244905254776834002228683580, 5.58488565043829121858857465160, 5.97175004030071896383000131913, 6.11779289366797645606845046099, 6.40870525303038711965989394035, 7.00815450400771221949493473984, 7.32339905040966103274605126905, 7.978997333339432818276948388429, 8.104743937011490115203757494307, 8.463515552407996208283127995058, 8.712607036135152781559145359729

Graph of the ZZ-function along the critical line