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Properties

Label 2-182-13.10-c1-0-2
Degree $2$
Conductor $182$
Sign $0.986 - 0.166i$
Analytic cond. $1.45327$
Root an. cond. $1.20551$
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  + (0.866 − 0.5i)2-s + (0.233 + 0.404i)3-s + (0.499 − 0.866i)4-s + 3.38i·5-s + (0.404 + 0.233i)6-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (1.39 − 2.40i)9-s + (1.69 + 2.93i)10-s + (0.712 − 0.411i)11-s + 0.466·12-s + (−2.74 + 2.33i)13-s + 0.999·14-s + (−1.36 + 0.790i)15-s + (−0.5 − 0.866i)16-s + (2.29 − 3.96i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.134 + 0.233i)3-s + (0.249 − 0.433i)4-s + 1.51i·5-s + (0.164 + 0.0952i)6-s + (0.327 + 0.188i)7-s − 0.353i·8-s + (0.463 − 0.803i)9-s + (0.535 + 0.928i)10-s + (0.214 − 0.124i)11-s + 0.134·12-s + (−0.762 + 0.646i)13-s + 0.267·14-s + (−0.353 + 0.204i)15-s + (−0.125 − 0.216i)16-s + (0.555 − 0.962i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182\)    =    \(2 \cdot 7 \cdot 13\)
Sign: $0.986 - 0.166i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{182} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 182,\ (\ :1/2),\ 0.986 - 0.166i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68242 + 0.140753i\)
\(L(\frac12)\) \(\approx\) \(1.68242 + 0.140753i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (2.74 - 2.33i)T \)
good3 \( 1 + (-0.233 - 0.404i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3.38iT - 5T^{2} \)
11 \( 1 + (-0.712 + 0.411i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.29 + 3.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.11 + 2.95i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.06 + 5.30i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.43 - 5.94i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.28iT - 31T^{2} \)
37 \( 1 + (8.39 - 4.84i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.0774 - 0.0446i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.67 + 6.36i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 11.1iT - 47T^{2} \)
53 \( 1 + 7.01T + 53T^{2} \)
59 \( 1 + (-1.50 - 0.870i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.18 - 2.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.252 + 0.145i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.48 - 5.47i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 - 9.95T + 79T^{2} \)
83 \( 1 + 3.23iT - 83T^{2} \)
89 \( 1 + (-6.96 + 4.02i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.7 - 7.38i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.43149830300066034502478869166, −11.78428808335274544425569796890, −10.67823286160285784151417173151, −10.04437342506357320494766406373, −8.801020336925080270636024065022, −6.99682539185956677201504003268, −6.59751808707102579979074398314, −4.89064260160661478474248865315, −3.58666847824751589471049600731, −2.42201573940902990249638205155, 1.81819853738505224793291464958, 4.08189916177612932704119178990, 4.97311401959823166801224759110, 6.04033041721671168800926284826, 7.82951216580419489490577584180, 8.068564251247486648808607374979, 9.530632497214743856930428717502, 10.69439519026765816267062482886, 12.18521114961169340473950518398, 12.63271699982723686166685513294

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