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Properties

Label 2-185-185.8-c1-0-7
Degree $2$
Conductor $185$
Sign $0.864 - 0.502i$
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  + (0.971 − 0.560i)2-s + (0.168 + 0.628i)3-s + (−0.371 + 0.642i)4-s + (0.550 + 2.16i)5-s + (0.516 + 0.516i)6-s + (−0.377 − 1.40i)7-s + 3.07i·8-s + (2.23 − 1.28i)9-s + (1.74 + 1.79i)10-s + 0.304i·11-s + (−0.466 − 0.125i)12-s + (−0.187 − 0.108i)13-s + (−1.15 − 1.15i)14-s + (−1.27 + 0.711i)15-s + (0.982 + 1.70i)16-s + (0.836 + 1.44i)17-s + ⋯
L(s)  = 1  + (0.686 − 0.396i)2-s + (0.0972 + 0.363i)3-s + (−0.185 + 0.321i)4-s + (0.246 + 0.969i)5-s + (0.210 + 0.210i)6-s + (−0.142 − 0.532i)7-s + 1.08i·8-s + (0.743 − 0.429i)9-s + (0.553 + 0.567i)10-s + 0.0918i·11-s + (−0.134 − 0.0361i)12-s + (−0.0520 − 0.0300i)13-s + (−0.308 − 0.308i)14-s + (−0.327 + 0.183i)15-s + (0.245 + 0.425i)16-s + (0.202 + 0.351i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 - 0.502i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 - 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.864 - 0.502i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ 0.864 - 0.502i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57738 + 0.425284i\)
\(L(\frac12)\) \(\approx\) \(1.57738 + 0.425284i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.550 - 2.16i)T \)
37 \( 1 + (6.04 - 0.671i)T \)
good2 \( 1 + (-0.971 + 0.560i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.168 - 0.628i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.377 + 1.40i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 - 0.304iT - 11T^{2} \)
13 \( 1 + (0.187 + 0.108i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.836 - 1.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.91 + 7.14i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 5.69iT - 23T^{2} \)
29 \( 1 + (1.05 + 1.05i)T + 29iT^{2} \)
31 \( 1 + (-2.64 + 2.64i)T - 31iT^{2} \)
41 \( 1 + (-2.07 - 1.19i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 7.13iT - 43T^{2} \)
47 \( 1 + (3.92 - 3.92i)T - 47iT^{2} \)
53 \( 1 + (-2.13 + 7.98i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-6.39 - 1.71i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.09 - 7.81i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (7.56 - 2.02i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-5.55 + 9.62i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.42 - 8.42i)T - 73iT^{2} \)
79 \( 1 + (1.51 + 5.66i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-1.06 + 3.95i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-0.842 + 3.14i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + 4.65T + 97T^{2} \)
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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

−12.90364816873551118441760728142, −11.73060127735916032805059890374, −10.77970005151738748840887127308, −9.976049065314326583036655278351, −8.770663367840765287241051232704, −7.39002460635785199731029584500, −6.41533401691973288787377853463, −4.73902872633147418417028453268, −3.79155902920614738420095918507, −2.60802192733188206515852863827, 1.59602241484419553616934355329, 3.92542510759121425279618667789, 5.15298205861433889567687448615, 5.89959206909677385396372585302, 7.21414648288229614498057017328, 8.440262732687310625284943253344, 9.544259502346980451085106765391, 10.34407896553285952339079085615, 12.10789130378117002614682499004, 12.63856793610185335353984345097

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