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Properties

Label 4-245e2-1.1-c3e2-0-12
Degree 44
Conductor 6002560025
Sign 11
Analytic cond. 208.960208.960
Root an. cond. 3.802033.80203
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 2·3-s + 8·4-s + 5·5-s + 6·6-s − 45·8-s + 27·9-s − 15·10-s + 45·11-s − 16·12-s − 118·13-s − 10·15-s + 135·16-s − 54·17-s − 81·18-s − 121·19-s + 40·20-s − 135·22-s − 69·23-s + 90·24-s + 354·26-s − 154·27-s − 324·29-s + 30·30-s − 88·31-s − 360·32-s − 90·33-s + ⋯
L(s)  = 1  − 1.06·2-s − 0.384·3-s + 4-s + 0.447·5-s + 0.408·6-s − 1.98·8-s + 9-s − 0.474·10-s + 1.23·11-s − 0.384·12-s − 2.51·13-s − 0.172·15-s + 2.10·16-s − 0.770·17-s − 1.06·18-s − 1.46·19-s + 0.447·20-s − 1.30·22-s − 0.625·23-s + 0.765·24-s + 2.67·26-s − 1.09·27-s − 2.07·29-s + 0.182·30-s − 0.509·31-s − 1.98·32-s − 0.474·33-s + ⋯

Functional equation

Λ(s)=(60025s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(60025s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 6002560025    =    52⋅745^{2} \cdot 7^{4}
Sign: 11
Analytic conductor: 208.960208.960
Root analytic conductor: 3.802033.80203
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 60025, ( :3/2,3/2), 1)(4,\ 60025,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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)
bad5C2C_2 1−pT+p2T2 1 - p T + p^{2} T^{2}
7 1 1
good2C22C_2^2 1+3T+T2+3p3T3+p6T4 1 + 3 T + T^{2} + 3 p^{3} T^{3} + p^{6} T^{4}
3C22C_2^2 1+2T−23T2+2p3T3+p6T4 1 + 2 T - 23 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4}
11C22C_2^2 1−45T+694T2−45p3T3+p6T4 1 - 45 T + 694 T^{2} - 45 p^{3} T^{3} + p^{6} T^{4}
13C2C_2 (1+59T+p3T2)2 ( 1 + 59 T + p^{3} T^{2} )^{2}
17C22C_2^2 1+54T−1997T2+54p3T3+p6T4 1 + 54 T - 1997 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4}
19C22C_2^2 1+121T+7782T2+121p3T3+p6T4 1 + 121 T + 7782 T^{2} + 121 p^{3} T^{3} + p^{6} T^{4}
23C22C_2^2 1+3pT−14p2T2+3p4T3+p6T4 1 + 3 p T - 14 p^{2} T^{2} + 3 p^{4} T^{3} + p^{6} T^{4}
29C2C_2 (1+162T+p3T2)2 ( 1 + 162 T + p^{3} T^{2} )^{2}
31C22C_2^2 1+88T−22047T2+88p3T3+p6T4 1 + 88 T - 22047 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4}
37C22C_2^2 1−7pT+12p2T2−7p4T3+p6T4 1 - 7 p T + 12 p^{2} T^{2} - 7 p^{4} T^{3} + p^{6} T^{4}
41C2C_2 (1+195T+p3T2)2 ( 1 + 195 T + p^{3} T^{2} )^{2}
43C2C_2 (1+286T+p3T2)2 ( 1 + 286 T + p^{3} T^{2} )^{2}
47C22C_2^2 1−45T−101798T2−45p3T3+p6T4 1 - 45 T - 101798 T^{2} - 45 p^{3} T^{3} + p^{6} T^{4}
53C22C_2^2 1+597T+207532T2+597p3T3+p6T4 1 + 597 T + 207532 T^{2} + 597 p^{3} T^{3} + p^{6} T^{4}
59C22C_2^2 1+360T−75779T2+360p3T3+p6T4 1 + 360 T - 75779 T^{2} + 360 p^{3} T^{3} + p^{6} T^{4}
61C22C_2^2 1−392T−73317T2−392p3T3+p6T4 1 - 392 T - 73317 T^{2} - 392 p^{3} T^{3} + p^{6} T^{4}
67C22C_2^2 1−280T−222363T2−280p3T3+p6T4 1 - 280 T - 222363 T^{2} - 280 p^{3} T^{3} + p^{6} T^{4}
71C2C_2 (1−48T+p3T2)2 ( 1 - 48 T + p^{3} T^{2} )^{2}
73C22C_2^2 1−668T+57207T2−668p3T3+p6T4 1 - 668 T + 57207 T^{2} - 668 p^{3} T^{3} + p^{6} T^{4}
79C22C_2^2 1+782T+118485T2+782p3T3+p6T4 1 + 782 T + 118485 T^{2} + 782 p^{3} T^{3} + p^{6} T^{4}
83C2C_2 (1+768T+p3T2)2 ( 1 + 768 T + p^{3} T^{2} )^{2}
89C22C_2^2 1+1194T+720667T2+1194p3T3+p6T4 1 + 1194 T + 720667 T^{2} + 1194 p^{3} T^{3} + p^{6} T^{4}
97C2C_2 (1+902T+p3T2)2 ( 1 + 902 T + p^{3} T^{2} )^{2}
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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

−11.58267209736287399651128661448, −10.92346992168105237331954304574, −10.26021769626393133744860111608, −9.672811118446763062863025039888, −9.542129607324851715494398006041, −9.381113582207653616508340074642, −8.428577478688992043387629518628, −8.148237290720854196389580006949, −7.21325014567227958757266482476, −6.95935213476336409337267378419, −6.58319668320953836951187300449, −5.88957691589284154573896068572, −5.39143264305443359615586135431, −4.52165325647870607520955010196, −3.92838727740324194656104250926, −2.97775683614366429770859189630, −1.95687392609680384929042194330, −1.80984154891676207836866886978, 0, 0, 1.80984154891676207836866886978, 1.95687392609680384929042194330, 2.97775683614366429770859189630, 3.92838727740324194656104250926, 4.52165325647870607520955010196, 5.39143264305443359615586135431, 5.88957691589284154573896068572, 6.58319668320953836951187300449, 6.95935213476336409337267378419, 7.21325014567227958757266482476, 8.148237290720854196389580006949, 8.428577478688992043387629518628, 9.381113582207653616508340074642, 9.542129607324851715494398006041, 9.672811118446763062863025039888, 10.26021769626393133744860111608, 10.92346992168105237331954304574, 11.58267209736287399651128661448

Graph of the ZZ-function along the critical line