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Properties

Label 2-8325-1.1-c1-0-239
Degree 22
Conductor 83258325
Sign −1-1
Analytic cond. 66.475466.4754
Root an. cond. 8.153248.15324
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s + 7-s + 5·11-s + 2·13-s − 2·14-s − 4·16-s − 10·22-s + 2·23-s − 4·26-s + 2·28-s − 6·29-s − 4·31-s + 8·32-s + 37-s + 9·41-s − 2·43-s + 10·44-s − 4·46-s − 9·47-s − 6·49-s + 4·52-s + 53-s + 12·58-s − 8·59-s − 8·61-s + 8·62-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.377·7-s + 1.50·11-s + 0.554·13-s − 0.534·14-s − 16-s − 2.13·22-s + 0.417·23-s − 0.784·26-s + 0.377·28-s − 1.11·29-s − 0.718·31-s + 1.41·32-s + 0.164·37-s + 1.40·41-s − 0.304·43-s + 1.50·44-s − 0.589·46-s − 1.31·47-s − 6/7·49-s + 0.554·52-s + 0.137·53-s + 1.57·58-s − 1.04·59-s − 1.02·61-s + 1.01·62-s + ⋯

Functional equation

Λ(s)=(8325s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(8325s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 83258325    =    32⋅52⋅373^{2} \cdot 5^{2} \cdot 37
Sign: −1-1
Analytic conductor: 66.475466.4754
Root analytic conductor: 8.153248.15324
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 8325, ( :1/2), −1)(2,\ 8325,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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}
ppFp(T)F_p(T)
bad3 1 1
5 1 1
37 1−T 1 - T
good2 1+pT+pT2 1 + p T + p T^{2}
7 1−T+pT2 1 - T + p T^{2}
11 1−5T+pT2 1 - 5 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+pT2 1 + p T^{2}
23 1−2T+pT2 1 - 2 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
41 1−9T+pT2 1 - 9 T + p T^{2}
43 1+2T+pT2 1 + 2 T + p T^{2}
47 1+9T+pT2 1 + 9 T + p T^{2}
53 1−T+pT2 1 - T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1+9T+pT2 1 + 9 T + p T^{2}
73 1−T+pT2 1 - T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1+15T+pT2 1 + 15 T + p T^{2}
89 1+4T+pT2 1 + 4 T + p T^{2}
97 1+4T+pT2 1 + 4 T + p T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.65121136122522161197318054345, −6.94384664150198731468349379760, −6.36962640367681528657535462564, −5.55465832553444053493877306394, −4.50549554166763456403503901204, −3.91879626410547859574461452178, −2.89439405381897961095418638007, −1.61293502722827766917172111896, −1.35033591033027974757425948725, 0, 1.35033591033027974757425948725, 1.61293502722827766917172111896, 2.89439405381897961095418638007, 3.91879626410547859574461452178, 4.50549554166763456403503901204, 5.55465832553444053493877306394, 6.36962640367681528657535462564, 6.94384664150198731468349379760, 7.65121136122522161197318054345

Graph of the ZZ-function along the critical line