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Properties

Label 2-82-41.14-c4-0-12
Degree $2$
Conductor $82$
Sign $-0.907 - 0.420i$
Analytic cond. $8.47633$
Root an. cond. $2.91141$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 2i)2-s + (−3.50 − 8.45i)3-s + 8i·4-s + (13.5 − 13.5i)5-s + (−9.90 + 23.9i)6-s + (−23.8 − 57.5i)7-s + (16 − 16i)8-s + (−1.88 + 1.88i)9-s − 54.3·10-s + (−20.9 + 8.69i)11-s + (67.6 − 28.0i)12-s + (−31.3 − 75.7i)13-s + (−67.3 + 162. i)14-s + (−162. − 67.2i)15-s − 64·16-s + (−173. + 420. i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.388 − 0.938i)3-s + 0.5i·4-s + (0.543 − 0.543i)5-s + (−0.275 + 0.663i)6-s + (−0.486 − 1.17i)7-s + (0.250 − 0.250i)8-s + (−0.0233 + 0.0233i)9-s − 0.543·10-s + (−0.173 + 0.0718i)11-s + (0.469 − 0.194i)12-s + (−0.185 − 0.447i)13-s + (−0.343 + 0.829i)14-s + (−0.721 − 0.298i)15-s − 0.250·16-s + (−0.602 + 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(82\)    =    \(2 \cdot 41\)
Sign: $-0.907 - 0.420i$
Analytic conductor: \(8.47633\)
Root analytic conductor: \(2.91141\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{82} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 82,\ (\ :2),\ -0.907 - 0.420i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.163702 + 0.741579i\)
\(L(\frac12)\) \(\approx\) \(0.163702 + 0.741579i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 + 2i)T \)
41 \( 1 + (1.61e3 - 457. i)T \)
good3 \( 1 + (3.50 + 8.45i)T + (-57.2 + 57.2i)T^{2} \)
5 \( 1 + (-13.5 + 13.5i)T - 625iT^{2} \)
7 \( 1 + (23.8 + 57.5i)T + (-1.69e3 + 1.69e3i)T^{2} \)
11 \( 1 + (20.9 - 8.69i)T + (1.03e4 - 1.03e4i)T^{2} \)
13 \( 1 + (31.3 + 75.7i)T + (-2.01e4 + 2.01e4i)T^{2} \)
17 \( 1 + (173. - 420. i)T + (-5.90e4 - 5.90e4i)T^{2} \)
19 \( 1 + (114. - 275. i)T + (-9.21e4 - 9.21e4i)T^{2} \)
23 \( 1 + 365. iT - 2.79e5T^{2} \)
29 \( 1 + (224. + 541. i)T + (-5.00e5 + 5.00e5i)T^{2} \)
31 \( 1 + 429. iT - 9.23e5T^{2} \)
37 \( 1 - 1.39e3T + 1.87e6T^{2} \)
43 \( 1 + (157. + 157. i)T + 3.41e6iT^{2} \)
47 \( 1 + (-1.60e3 + 3.86e3i)T + (-3.45e6 - 3.45e6i)T^{2} \)
53 \( 1 + (-3.05e3 + 1.26e3i)T + (5.57e6 - 5.57e6i)T^{2} \)
59 \( 1 + 1.14e3T + 1.21e7T^{2} \)
61 \( 1 + (-31.8 - 31.8i)T + 1.38e7iT^{2} \)
67 \( 1 + (831. - 2.00e3i)T + (-1.42e7 - 1.42e7i)T^{2} \)
71 \( 1 + (2.40e3 + 5.81e3i)T + (-1.79e7 + 1.79e7i)T^{2} \)
73 \( 1 + (1.56e3 + 1.56e3i)T + 2.83e7iT^{2} \)
79 \( 1 + (4.74e3 - 1.96e3i)T + (2.75e7 - 2.75e7i)T^{2} \)
83 \( 1 - 709.T + 4.74e7T^{2} \)
89 \( 1 + (786. + 1.89e3i)T + (-4.43e7 + 4.43e7i)T^{2} \)
97 \( 1 + (-1.21e4 - 5.03e3i)T + (6.25e7 + 6.25e7i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.11543724528571708775497894939, −12.04559319397503286161409214850, −10.63037437899060639775086048662, −9.845362013839556288374624440321, −8.357839268025003640079417366178, −7.18563580095063412291424118723, −6.04076002914725765832945880247, −3.99681159947776394973821805503, −1.78220600468556040006204173754, −0.45737260428299239425265973133, 2.56872399439332084458299122792, 4.81403811220777601169596421069, 5.92299882894475692593985944528, 7.13823810528419223941898749135, 8.997519869756915017649470667442, 9.602688426853818273827944993629, 10.68972697273821505657395244293, 11.71847163651593333992907194017, 13.31112324531674402601313458681, 14.50133111329165708117718954930

Graph of the $Z$-function along the critical line