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Properties

Label 2-832-1.1-c3-0-60
Degree 22
Conductor 832832
Sign −1-1
Analytic cond. 49.089549.0895
Root an. cond. 7.006397.00639
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.68·3-s − 0.561·5-s + 18.1·7-s − 13.4·9-s − 64.7·11-s + 13·13-s − 2.06·15-s − 25.5·17-s + 107.·19-s + 66.9·21-s + 73.2·23-s − 124.·25-s − 148.·27-s − 175.·29-s − 113.·31-s − 238.·33-s − 10.2·35-s − 114.·37-s + 47.9·39-s − 69.6·41-s − 438.·43-s + 7.53·45-s − 31.9·47-s − 12.5·49-s − 94.1·51-s − 2.84·53-s + 36.3·55-s + ⋯
L(s)  = 1  + 0.709·3-s − 0.0502·5-s + 0.981·7-s − 0.497·9-s − 1.77·11-s + 0.277·13-s − 0.0356·15-s − 0.364·17-s + 1.30·19-s + 0.695·21-s + 0.664·23-s − 0.997·25-s − 1.06·27-s − 1.12·29-s − 0.655·31-s − 1.25·33-s − 0.0492·35-s − 0.510·37-s + 0.196·39-s − 0.265·41-s − 1.55·43-s + 0.0249·45-s − 0.0991·47-s − 0.0367·49-s − 0.258·51-s − 0.00737·53-s + 0.0891·55-s + ⋯

Functional equation

Λ(s)=(832s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(832s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 832832    =    26⋅132^{6} \cdot 13
Sign: −1-1
Analytic conductor: 49.089549.0895
Root analytic conductor: 7.006397.00639
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 832, ( :3/2), −1)(2,\ 832,\ (\ :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}
ppFp(T)F_p(T)
bad2 1 1
13 1−13T 1 - 13T
good3 1−3.68T+27T2 1 - 3.68T + 27T^{2}
5 1+0.561T+125T2 1 + 0.561T + 125T^{2}
7 1−18.1T+343T2 1 - 18.1T + 343T^{2}
11 1+64.7T+1.33e3T2 1 + 64.7T + 1.33e3T^{2}
17 1+25.5T+4.91e3T2 1 + 25.5T + 4.91e3T^{2}
19 1−107.T+6.85e3T2 1 - 107.T + 6.85e3T^{2}
23 1−73.2T+1.21e4T2 1 - 73.2T + 1.21e4T^{2}
29 1+175.T+2.43e4T2 1 + 175.T + 2.43e4T^{2}
31 1+113.T+2.97e4T2 1 + 113.T + 2.97e4T^{2}
37 1+114.T+5.06e4T2 1 + 114.T + 5.06e4T^{2}
41 1+69.6T+6.89e4T2 1 + 69.6T + 6.89e4T^{2}
43 1+438.T+7.95e4T2 1 + 438.T + 7.95e4T^{2}
47 1+31.9T+1.03e5T2 1 + 31.9T + 1.03e5T^{2}
53 1+2.84T+1.48e5T2 1 + 2.84T + 1.48e5T^{2}
59 1+71.6T+2.05e5T2 1 + 71.6T + 2.05e5T^{2}
61 1−920.T+2.26e5T2 1 - 920.T + 2.26e5T^{2}
67 1−444.T+3.00e5T2 1 - 444.T + 3.00e5T^{2}
71 1+541.T+3.57e5T2 1 + 541.T + 3.57e5T^{2}
73 1−764.T+3.89e5T2 1 - 764.T + 3.89e5T^{2}
79 1+421.T+4.93e5T2 1 + 421.T + 4.93e5T^{2}
83 1+603.T+5.71e5T2 1 + 603.T + 5.71e5T^{2}
89 1+1.15e3T+7.04e5T2 1 + 1.15e3T + 7.04e5T^{2}
97 1−583.T+9.12e5T2 1 - 583.T + 9.12e5T^{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

−9.319499284172354490094845639702, −8.331002260396050721010931764949, −7.935250529032507360428747065525, −7.11145528861906989029965312862, −5.50580407107914811657684156438, −5.16052919469470520776141001360, −3.70945114684107574543167233522, −2.74366042504932949051645542020, −1.74628020280605270243765824030, 0, 1.74628020280605270243765824030, 2.74366042504932949051645542020, 3.70945114684107574543167233522, 5.16052919469470520776141001360, 5.50580407107914811657684156438, 7.11145528861906989029965312862, 7.935250529032507360428747065525, 8.331002260396050721010931764949, 9.319499284172354490094845639702

Graph of the ZZ-function along the critical line