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Properties

Label 2-912-1.1-c1-0-16
Degree 22
Conductor 912912
Sign −1-1
Analytic cond. 7.282357.28235
Root an. cond. 2.698582.69858
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  + 3-s − 3·5-s − 7-s + 9-s + 5·11-s − 6·13-s − 3·15-s − 5·17-s − 19-s − 21-s − 4·23-s + 4·25-s + 27-s + 6·29-s − 6·31-s + 5·33-s + 3·35-s − 8·37-s − 6·39-s − 8·41-s − 9·43-s − 3·45-s − 47-s − 6·49-s − 5·51-s + 2·53-s − 15·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s + 1.50·11-s − 1.66·13-s − 0.774·15-s − 1.21·17-s − 0.229·19-s − 0.218·21-s − 0.834·23-s + 4/5·25-s + 0.192·27-s + 1.11·29-s − 1.07·31-s + 0.870·33-s + 0.507·35-s − 1.31·37-s − 0.960·39-s − 1.24·41-s − 1.37·43-s − 0.447·45-s − 0.145·47-s − 6/7·49-s − 0.700·51-s + 0.274·53-s − 2.02·55-s + ⋯

Functional equation

Λ(s)=(912s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(912s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 912912    =    24⋅3⋅192^{4} \cdot 3 \cdot 19
Sign: −1-1
Analytic conductor: 7.282357.28235
Root analytic conductor: 2.698582.69858
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 912, ( :1/2), −1)(2,\ 912,\ (\ :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)
bad2 1 1
3 1−T 1 - T
19 1+T 1 + T
good5 1+3T+pT2 1 + 3 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+6T+pT2 1 + 6 T + p T^{2}
17 1+5T+pT2 1 + 5 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+6T+pT2 1 + 6 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1+8T+pT2 1 + 8 T + p T^{2}
43 1+9T+pT2 1 + 9 T + p T^{2}
47 1+T+pT2 1 + T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1−11T+pT2 1 - 11 T + p T^{2}
67 1+pT2 1 + p T^{2}
71 1−4T+pT2 1 - 4 T + p T^{2}
73 1+11T+pT2 1 + 11 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1−10T+pT2 1 - 10 T + p T^{2}
97 1+10T+pT2 1 + 10 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

−9.585874925160168888904155563723, −8.743979403432894067002867605875, −8.098723763596542718825097439727, −6.98995551256525595204221630124, −6.70923923756293259323170265958, −4.96063409732475955643392332129, −4.09803140790119357857110127232, −3.38752685173358042429193565276, −2.02959224696734609090119252606, 0, 2.02959224696734609090119252606, 3.38752685173358042429193565276, 4.09803140790119357857110127232, 4.96063409732475955643392332129, 6.70923923756293259323170265958, 6.98995551256525595204221630124, 8.098723763596542718825097439727, 8.743979403432894067002867605875, 9.585874925160168888904155563723

Graph of the ZZ-function along the critical line