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Properties

Label 2-185-185.103-c1-0-12
Degree $2$
Conductor $185$
Sign $0.805 + 0.592i$
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  + (0.659 + 1.14i)2-s + (−0.318 − 1.18i)3-s + (0.131 − 0.227i)4-s + (−1.74 − 1.39i)5-s + (1.14 − 1.14i)6-s + (−1.10 − 4.13i)7-s + 2.98·8-s + (1.28 − 0.743i)9-s + (0.438 − 2.91i)10-s + 4.89i·11-s + (−0.311 − 0.0835i)12-s + (−1.48 + 2.57i)13-s + (3.98 − 3.98i)14-s + (−1.09 + 2.52i)15-s + (1.70 + 2.94i)16-s + (3.65 − 2.10i)17-s + ⋯
L(s)  = 1  + (0.466 + 0.807i)2-s + (−0.183 − 0.686i)3-s + (0.0656 − 0.113i)4-s + (−0.781 − 0.623i)5-s + (0.468 − 0.468i)6-s + (−0.418 − 1.56i)7-s + 1.05·8-s + (0.429 − 0.247i)9-s + (0.138 − 0.921i)10-s + 1.47i·11-s + (−0.0900 − 0.0241i)12-s + (−0.412 + 0.714i)13-s + (1.06 − 1.06i)14-s + (−0.283 + 0.651i)15-s + (0.425 + 0.737i)16-s + (0.885 − 0.511i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.805 + 0.592i$
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: \((2,\ 185,\ (\ :1/2),\ 0.805 + 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27200 - 0.417770i\)
\(L(\frac12)\) \(\approx\) \(1.27200 - 0.417770i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.74 + 1.39i)T \)
37 \( 1 + (-6.07 + 0.219i)T \)
good2 \( 1 + (-0.659 - 1.14i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.318 + 1.18i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (1.10 + 4.13i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 - 4.89iT - 11T^{2} \)
13 \( 1 + (1.48 - 2.57i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.65 + 2.10i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.90 - 1.31i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 6.57T + 23T^{2} \)
29 \( 1 + (-0.417 + 0.417i)T - 29iT^{2} \)
31 \( 1 + (-3.03 - 3.03i)T + 31iT^{2} \)
41 \( 1 + (0.897 + 0.518i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + 3.17T + 43T^{2} \)
47 \( 1 + (4.44 - 4.44i)T - 47iT^{2} \)
53 \( 1 + (-1.71 + 6.40i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.57 - 5.86i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.68 + 0.450i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (11.5 - 3.09i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.78 - 3.09i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.50 - 2.50i)T - 73iT^{2} \)
79 \( 1 + (-3.97 + 1.06i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (0.688 - 2.57i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-11.4 - 3.07i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 - 10.2iT - 97T^{2} \)
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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

−12.79292337455554266979799454116, −11.81793236737154024286878352335, −10.49334076222948915162889873343, −9.611569927746793511281977180419, −7.78478492190687005231491508185, −7.14576655437778626156499217778, −6.64118189682660776242991744448, −4.80096345924791403560239377820, −4.13804513614405016749638972337, −1.26642138077388130402095380531, 2.72834735453754405313923008585, 3.50266526138239289568530532852, 4.89413600902537948929707885905, 6.17879443877468677527816019810, 7.73643909049110586473480117055, 8.728037055599069716157426189684, 10.16325038752640566634208261314, 10.94025962878033184957323045778, 11.63973890821596236015944319827, 12.55434136845242904469814893711

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