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Properties

Label 2-185-185.103-c1-0-14
Degree $2$
Conductor $185$
Sign $-0.936 - 0.350i$
Analytic cond. $1.47723$
Root an. cond. $1.21541$
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  + (−0.747 − 1.29i)2-s + (−0.579 − 2.16i)3-s + (−0.117 + 0.203i)4-s + (−0.664 + 2.13i)5-s + (−2.36 + 2.36i)6-s + (−0.962 − 3.59i)7-s − 2.63·8-s + (−1.74 + 1.00i)9-s + (3.26 − 0.734i)10-s + 0.0710i·11-s + (0.507 + 0.135i)12-s + (−0.360 + 0.624i)13-s + (−3.93 + 3.93i)14-s + (5.00 + 0.200i)15-s + (2.20 + 3.82i)16-s + (−2.10 + 1.21i)17-s + ⋯
L(s)  = 1  + (−0.528 − 0.915i)2-s + (−0.334 − 1.24i)3-s + (−0.0586 + 0.101i)4-s + (−0.297 + 0.954i)5-s + (−0.965 + 0.965i)6-s + (−0.363 − 1.35i)7-s − 0.933·8-s + (−0.580 + 0.334i)9-s + (1.03 − 0.232i)10-s + 0.0214i·11-s + (0.146 + 0.0392i)12-s + (−0.0999 + 0.173i)13-s + (−1.05 + 1.05i)14-s + (1.29 + 0.0518i)15-s + (0.551 + 0.955i)16-s + (−0.509 + 0.294i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.936 - 0.350i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ -0.936 - 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.109318 + 0.604714i\)
\(L(\frac12)\) \(\approx\) \(0.109318 + 0.604714i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.664 - 2.13i)T \)
37 \( 1 + (-0.352 + 6.07i)T \)
good2 \( 1 + (0.747 + 1.29i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.579 + 2.16i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.962 + 3.59i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 - 0.0710iT - 11T^{2} \)
13 \( 1 + (0.360 - 0.624i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.10 - 1.21i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.55 + 0.415i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 5.82T + 23T^{2} \)
29 \( 1 + (-4.82 + 4.82i)T - 29iT^{2} \)
31 \( 1 + (6.73 + 6.73i)T + 31iT^{2} \)
41 \( 1 + (4.99 + 2.88i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 9.26T + 43T^{2} \)
47 \( 1 + (-1.36 + 1.36i)T - 47iT^{2} \)
53 \( 1 + (2.81 - 10.5i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.359 + 1.34i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-10.5 + 2.82i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (0.313 - 0.0839i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.252 - 0.438i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.91 + 3.91i)T - 73iT^{2} \)
79 \( 1 + (6.41 - 1.71i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-3.35 + 12.5i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-15.8 - 4.23i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 - 15.4iT - 97T^{2} \)
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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

−11.85147637280977392289157751224, −11.03638237388658131773071233217, −10.44024530180411921030824469777, −9.320590860788457212463928967629, −7.67398419004650568121631169674, −6.97497731993799421836329656238, −6.12492019762431460607885690994, −3.80222138568791120937673515120, −2.31777460483163660874347786954, −0.67717455472085654311900733371, 3.21448441206847729562508691653, 4.92118431274588682690269633571, 5.58778109626876504155877089516, 6.97713597727139687158017839485, 8.472572715796207125836258804912, 8.997481308010465701347107314677, 9.740004697140313932032829270002, 11.18108883559652852394290841232, 12.13648882544792194657328339250, 12.90102070610351067938804423174

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