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Properties

Label 2-185-185.97-c1-0-1
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: $\chi_{185} (97, \cdot )$
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

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

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