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Properties

Label 4-975e2-1.1-c1e2-0-29
Degree 44
Conductor 950625950625
Sign 11
Analytic cond. 60.612660.6126
Root an. cond. 2.790232.79023
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  + 3·2-s + 3-s + 4·4-s + 3·6-s + 3·8-s + 3·11-s + 4·12-s + 7·13-s + 3·16-s + 9·22-s + 3·23-s + 3·24-s + 21·26-s − 27-s + 6·32-s + 3·33-s + 3·37-s + 7·39-s + 6·41-s + 10·43-s + 12·44-s + 9·46-s + 3·48-s − 7·49-s + 28·52-s − 3·54-s − 18·59-s + ⋯
L(s)  = 1  + 2.12·2-s + 0.577·3-s + 2·4-s + 1.22·6-s + 1.06·8-s + 0.904·11-s + 1.15·12-s + 1.94·13-s + 3/4·16-s + 1.91·22-s + 0.625·23-s + 0.612·24-s + 4.11·26-s − 0.192·27-s + 1.06·32-s + 0.522·33-s + 0.493·37-s + 1.12·39-s + 0.937·41-s + 1.52·43-s + 1.80·44-s + 1.32·46-s + 0.433·48-s − 49-s + 3.88·52-s − 0.408·54-s − 2.34·59-s + ⋯

Functional equation

Λ(s)=(950625s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(950625s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 950625950625    =    32⋅54⋅1323^{2} \cdot 5^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 60.612660.6126
Root analytic conductor: 2.790232.79023
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 950625, ( :1/2,1/2), 1)(4,\ 950625,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 9.6451213559.645121355
L(12)L(\frac12) ≈\approx 9.6451213559.645121355
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)
bad3C2C_2 1−T+T2 1 - T + T^{2}
5 1 1
13C2C_2 1−7T+pT2 1 - 7 T + p T^{2}
good2C22C_2^2 1−3T+5T2−3pT3+p2T4 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4}
7C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
11C22C_2^2 1−3T+14T2−3pT3+p2T4 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4}
17C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
19C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
23C22C_2^2 1−3T−14T2−3pT3+p2T4 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4}
29C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
31C22C_2^2 1−14T2+p2T4 1 - 14 T^{2} + p^{2} T^{4}
37C22C_2^2 1−3T+40T2−3pT3+p2T4 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4}
41C22C_2^2 1−6T+53T2−6pT3+p2T4 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4}
43C22C_2^2 1−10T+57T2−10pT3+p2T4 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
53C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
59C22C_2^2 1+18T+167T2+18pT3+p2T4 1 + 18 T + 167 T^{2} + 18 p T^{3} + p^{2} T^{4}
61C22C_2^2 1+7T−12T2+7pT3+p2T4 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4}
67C2C_2 (1−11T+pT2)(1+5T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} )
71C22C_2^2 1+15T+146T2+15pT3+p2T4 1 + 15 T + 146 T^{2} + 15 p T^{3} + p^{2} T^{4}
73C22C_2^2 1−71T2+p2T4 1 - 71 T^{2} + p^{2} T^{4}
79C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
83C22C_2^2 1−91T2+p2T4 1 - 91 T^{2} + p^{2} T^{4}
89C22C_2^2 1+12T+137T2+12pT3+p2T4 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4}
97C22C_2^2 1−15T+172T2−15pT3+p2T4 1 - 15 T + 172 T^{2} - 15 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

−10.33655290533496836420308194191, −9.843963231527390580171107357540, −9.105843186413900524784805238835, −9.069084932299466622369564730209, −8.639088792803456785164918736586, −8.046985918205892511596777274801, −7.55712781602263027829023812591, −7.26193385574861545151219010435, −6.31859453965618835958929932457, −6.30789815043422501137767938775, −5.88799104523779201314772109017, −5.46145510037921279529980404261, −4.65839595865025108296362679525, −4.52294054175957135050892081689, −3.88097013161846168618880854513, −3.70863208252683331751725555543, −3.01826139467454103950154576191, −2.79519202909103534337746912453, −1.67050637935550119428375772502, −1.15474611137418329177116412373, 1.15474611137418329177116412373, 1.67050637935550119428375772502, 2.79519202909103534337746912453, 3.01826139467454103950154576191, 3.70863208252683331751725555543, 3.88097013161846168618880854513, 4.52294054175957135050892081689, 4.65839595865025108296362679525, 5.46145510037921279529980404261, 5.88799104523779201314772109017, 6.30789815043422501137767938775, 6.31859453965618835958929932457, 7.26193385574861545151219010435, 7.55712781602263027829023812591, 8.046985918205892511596777274801, 8.639088792803456785164918736586, 9.069084932299466622369564730209, 9.105843186413900524784805238835, 9.843963231527390580171107357540, 10.33655290533496836420308194191

Graph of the ZZ-function along the critical line