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Properties

Label 2-968-1.1-c1-0-10
Degree 22
Conductor 968968
Sign 11
Analytic cond. 7.729517.72951
Root an. cond. 2.780202.78020
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 4·7-s − 2·9-s + 4·13-s + 15-s + 4·17-s − 4·19-s + 4·21-s − 3·23-s − 4·25-s − 5·27-s + 8·29-s + 9·31-s + 4·35-s − 5·37-s + 4·39-s − 12·41-s + 8·43-s − 2·45-s + 4·47-s + 9·49-s + 4·51-s − 10·53-s − 4·57-s + 7·59-s − 8·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.51·7-s − 2/3·9-s + 1.10·13-s + 0.258·15-s + 0.970·17-s − 0.917·19-s + 0.872·21-s − 0.625·23-s − 4/5·25-s − 0.962·27-s + 1.48·29-s + 1.61·31-s + 0.676·35-s − 0.821·37-s + 0.640·39-s − 1.87·41-s + 1.21·43-s − 0.298·45-s + 0.583·47-s + 9/7·49-s + 0.560·51-s − 1.37·53-s − 0.529·57-s + 0.911·59-s − 1.02·61-s + ⋯

Functional equation

Λ(s)=(968s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(968s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 968968    =    23⋅1122^{3} \cdot 11^{2}
Sign: 11
Analytic conductor: 7.729517.72951
Root analytic conductor: 2.780202.78020
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 968, ( :1/2), 1)(2,\ 968,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.4087561042.408756104
L(12)L(\frac12) ≈\approx 2.4087561042.408756104
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
11 1 1
good3 1−T+pT2 1 - T + p T^{2}
5 1−T+pT2 1 - T + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1−4T+pT2 1 - 4 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+3T+pT2 1 + 3 T + p T^{2}
29 1−8T+pT2 1 - 8 T + p T^{2}
31 1−9T+pT2 1 - 9 T + p T^{2}
37 1+5T+pT2 1 + 5 T + p T^{2}
41 1+12T+pT2 1 + 12 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1−7T+pT2 1 - 7 T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1−11T+pT2 1 - 11 T + p T^{2}
71 1+9T+pT2 1 + 9 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+T+pT2 1 + T + p T^{2}
97 1−T+pT2 1 - 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

−10.09251216461769686180120999755, −8.955204884515233663001449782687, −8.185775398334261426246211010655, −8.013124707654302294842241987618, −6.50037997902893774125269658524, −5.69647785290924362492290860490, −4.74414023623583900243885089454, −3.66738310623365329492205537138, −2.44137744204399593498082489636, −1.39420752604704988889459239823, 1.39420752604704988889459239823, 2.44137744204399593498082489636, 3.66738310623365329492205537138, 4.74414023623583900243885089454, 5.69647785290924362492290860490, 6.50037997902893774125269658524, 8.013124707654302294842241987618, 8.185775398334261426246211010655, 8.955204884515233663001449782687, 10.09251216461769686180120999755

Graph of the ZZ-function along the critical line