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Properties

Label 2-82-41.23-c1-0-1
Degree $2$
Conductor $82$
Sign $0.119 - 0.992i$
Analytic cond. $0.654773$
Root an. cond. $0.809180$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)2-s + 1.49i·3-s + (−0.809 + 0.587i)4-s + (0.5 − 0.363i)5-s + (−1.42 + 0.462i)6-s + (−1.42 − 0.462i)7-s + (−0.809 − 0.587i)8-s + 0.759·9-s + (0.5 + 0.363i)10-s + (0.543 − 0.748i)11-s + (−0.879 − 1.21i)12-s + (1.18 − 0.385i)13-s − 1.49i·14-s + (0.543 + 0.748i)15-s + (0.309 − 0.951i)16-s + (3.08 − 4.24i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + 0.864i·3-s + (−0.404 + 0.293i)4-s + (0.223 − 0.162i)5-s + (−0.581 + 0.188i)6-s + (−0.538 − 0.174i)7-s + (−0.286 − 0.207i)8-s + 0.253·9-s + (0.158 + 0.114i)10-s + (0.163 − 0.225i)11-s + (−0.253 − 0.349i)12-s + (0.328 − 0.106i)13-s − 0.400i·14-s + (0.140 + 0.193i)15-s + (0.0772 − 0.237i)16-s + (0.748 − 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.119 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.119 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(82\)    =    \(2 \cdot 41\)
Sign: $0.119 - 0.992i$
Analytic conductor: \(0.654773\)
Root analytic conductor: \(0.809180\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{82} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 82,\ (\ :1/2),\ 0.119 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.772708 + 0.685245i\)
\(L(\frac12)\) \(\approx\) \(0.772708 + 0.685245i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (-6.06 + 2.04i)T \)
good3 \( 1 - 1.49iT - 3T^{2} \)
5 \( 1 + (-0.5 + 0.363i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.42 + 0.462i)T + (5.66 + 4.11i)T^{2} \)
11 \( 1 + (-0.543 + 0.748i)T + (-3.39 - 10.4i)T^{2} \)
13 \( 1 + (-1.18 + 0.385i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-3.08 + 4.24i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (3.72 + 1.21i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.220 + 0.677i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (5.01 + 6.90i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.42 + 1.03i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (7.33 - 5.33i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (-3.75 - 11.5i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (7.23 - 2.35i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.89 - 3.98i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.900 + 2.76i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.515 + 1.58i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (1.67 + 2.30i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-7.52 + 10.3i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 - 6.70T + 73T^{2} \)
79 \( 1 + 4.70iT - 79T^{2} \)
83 \( 1 - 4.06T + 83T^{2} \)
89 \( 1 + (8.19 + 2.66i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (-2.35 - 3.24i)T + (-29.9 + 92.2i)T^{2} \)
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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

−14.78884949580427285564024997706, −13.61419269619582565759697128073, −12.74271936586549515009053231474, −11.23335781819644149792505209228, −9.890418888042884644703780443161, −9.175194905862104358706880634617, −7.64254697640446582905488348459, −6.25305171236918178091195718531, −4.90672396774604335239239652214, −3.57547498264736604828104940000, 1.88520175156583660627649230278, 3.81035000538076172193932523219, 5.81349133720908768291999327942, 6.99109430447187867208959541816, 8.487478138101990457543217134709, 9.857254317696063254315043837028, 10.84681838547352176046137908249, 12.40190638078248471408961992598, 12.67775904949432291245565373191, 13.84165593409472411660658710143

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