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Properties

Label 2-82-41.3-c4-0-5
Degree $2$
Conductor $82$
Sign $0.967 + 0.254i$
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 + (−6.17 + 14.9i)3-s − 8i·4-s + (−15.3 − 15.3i)5-s + (−17.4 − 42.1i)6-s + (22.0 − 53.2i)7-s + (16 + 16i)8-s + (−127. − 127. i)9-s + 61.4·10-s + (83.2 + 34.4i)11-s + (119. + 49.4i)12-s + (−67.8 + 163. i)13-s + (62.3 + 150. i)14-s + (323. − 134. i)15-s − 64·16-s + (−126. − 304. i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.686 + 1.65i)3-s − 0.5i·4-s + (−0.614 − 0.614i)5-s + (−0.485 − 1.17i)6-s + (0.450 − 1.08i)7-s + (0.250 + 0.250i)8-s + (−1.56 − 1.56i)9-s + 0.614·10-s + (0.688 + 0.285i)11-s + (0.828 + 0.343i)12-s + (−0.401 + 0.969i)13-s + (0.318 + 0.768i)14-s + (1.43 − 0.596i)15-s − 0.250·16-s + (−0.436 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(82\)    =    \(2 \cdot 41\)
Sign: $0.967 + 0.254i$
Analytic conductor: \(8.47633\)
Root analytic conductor: \(2.91141\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{82} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 82,\ (\ :2),\ 0.967 + 0.254i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.615584 - 0.0795977i\)
\(L(\frac12)\) \(\approx\) \(0.615584 - 0.0795977i\)
\(L(3)\) 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 \( 1 + (-953. - 1.38e3i)T \)
good3 \( 1 + (6.17 - 14.9i)T + (-57.2 - 57.2i)T^{2} \)
5 \( 1 + (15.3 + 15.3i)T + 625iT^{2} \)
7 \( 1 + (-22.0 + 53.2i)T + (-1.69e3 - 1.69e3i)T^{2} \)
11 \( 1 + (-83.2 - 34.4i)T + (1.03e4 + 1.03e4i)T^{2} \)
13 \( 1 + (67.8 - 163. i)T + (-2.01e4 - 2.01e4i)T^{2} \)
17 \( 1 + (126. + 304. i)T + (-5.90e4 + 5.90e4i)T^{2} \)
19 \( 1 + (-161. - 390. i)T + (-9.21e4 + 9.21e4i)T^{2} \)
23 \( 1 + 688. iT - 2.79e5T^{2} \)
29 \( 1 + (-438. + 1.05e3i)T + (-5.00e5 - 5.00e5i)T^{2} \)
31 \( 1 + 936. iT - 9.23e5T^{2} \)
37 \( 1 - 1.38e3T + 1.87e6T^{2} \)
43 \( 1 + (661. - 661. i)T - 3.41e6iT^{2} \)
47 \( 1 + (883. + 2.13e3i)T + (-3.45e6 + 3.45e6i)T^{2} \)
53 \( 1 + (2.18e3 + 904. i)T + (5.57e6 + 5.57e6i)T^{2} \)
59 \( 1 + 1.16e3T + 1.21e7T^{2} \)
61 \( 1 + (-548. + 548. i)T - 1.38e7iT^{2} \)
67 \( 1 + (3.12e3 + 7.55e3i)T + (-1.42e7 + 1.42e7i)T^{2} \)
71 \( 1 + (-3.36e3 + 8.12e3i)T + (-1.79e7 - 1.79e7i)T^{2} \)
73 \( 1 + (3.90e3 - 3.90e3i)T - 2.83e7iT^{2} \)
79 \( 1 + (-6.09e3 - 2.52e3i)T + (2.75e7 + 2.75e7i)T^{2} \)
83 \( 1 + 1.25e4T + 4.74e7T^{2} \)
89 \( 1 + (-860. + 2.07e3i)T + (-4.43e7 - 4.43e7i)T^{2} \)
97 \( 1 + (-7.77e3 + 3.22e3i)T + (6.25e7 - 6.25e7i)T^{2} \)
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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

−14.05674372939645893197331802677, −11.93181600649255194729316508933, −11.22313798827092850023020376800, −10.02320976205376155933441192845, −9.325921718971419751264982302543, −7.965228791352044602402285225356, −6.40131260821547222843566945838, −4.63615821516281101560103985483, −4.25575734064938796759873320220, −0.44179359734713768853262167436, 1.34031113203232200057142021312, 2.86789990411908094720263377791, 5.52724572439169439966590747133, 6.86376079496639583486571780000, 7.80707490061343685586052366870, 8.862295528290033623349770528325, 10.88419760120712679463890477077, 11.54515243372107047549321987317, 12.31810745559023580709513338769, 13.17496016778812887867099336915

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