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Properties

Label 2-82-41.18-c1-0-1
Degree $2$
Conductor $82$
Sign $0.723 - 0.690i$
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 + 0.381·3-s + (−0.809 + 0.587i)4-s + (2.11 − 1.53i)5-s + (0.118 + 0.363i)6-s + (−0.881 + 2.71i)7-s + (−0.809 − 0.587i)8-s − 2.85·9-s + (2.11 + 1.53i)10-s + (0.190 + 0.138i)11-s + (−0.309 + 0.224i)12-s + (−1.69 − 5.20i)13-s − 2.85·14-s + (0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (−0.190 − 0.138i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + 0.220·3-s + (−0.404 + 0.293i)4-s + (0.947 − 0.688i)5-s + (0.0481 + 0.148i)6-s + (−0.333 + 1.02i)7-s + (−0.286 − 0.207i)8-s − 0.951·9-s + (0.669 + 0.486i)10-s + (0.0575 + 0.0418i)11-s + (−0.0892 + 0.0648i)12-s + (−0.468 − 1.44i)13-s − 0.762·14-s + (0.208 − 0.151i)15-s + (0.0772 − 0.237i)16-s + (−0.0463 − 0.0336i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(82\)    =    \(2 \cdot 41\)
Sign: $0.723 - 0.690i$
Analytic conductor: \(0.654773\)
Root analytic conductor: \(0.809180\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{82} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 82,\ (\ :1/2),\ 0.723 - 0.690i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03766 + 0.415643i\)
\(L(\frac12)\) \(\approx\) \(1.03766 + 0.415643i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.30 - 3.57i)T \)
good3 \( 1 - 0.381T + 3T^{2} \)
5 \( 1 + (-2.11 + 1.53i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.881 - 2.71i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (-0.190 - 0.138i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (1.69 + 5.20i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.190 + 0.138i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.54 + 4.75i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-2.19 - 6.74i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (5.42 - 3.94i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (4.11 + 2.99i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.42 + 2.48i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (-3.14 - 9.68i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-0.763 - 2.35i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.78 + 5.65i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.97 - 6.06i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.69 + 8.28i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (2 - 1.45i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (8.16 + 5.93i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 - 1.23T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 - 2.61T + 83T^{2} \)
89 \( 1 + (-0.690 + 2.12i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (2.42 - 1.76i)T + (29.9 - 92.2i)T^{2} \)
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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

−14.61891671290607159684449526068, −13.28064746338124628564622109367, −12.80573844741939711638751349671, −11.36100940551440957985825314216, −9.493238558641730806570456864529, −8.992670370528146262243137628771, −7.63263162492437548654979636550, −5.78464837063909846041542472855, −5.33708315137934878523476586436, −2.86912089137386797881271569897, 2.35546353972840861196999447107, 3.97323508580289420812107875650, 5.84909988855053282759236017523, 7.10106203913230264904901001100, 8.926194525794375020162619209950, 9.988856200388498945585578804721, 10.82343167272197813708611258652, 11.98607956991596328680327732855, 13.36051921971141608299404250117, 14.19868526972989146868176547554

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