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Properties

Label 2-185-185.162-c1-0-1
Degree $2$
Conductor $185$
Sign $-0.986 - 0.161i$
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  + (0.407 + 0.235i)2-s + (−0.838 + 3.12i)3-s + (−0.889 − 1.54i)4-s + (−2.02 − 0.950i)5-s + (−1.07 + 1.07i)6-s + (−0.968 + 3.61i)7-s − 1.77i·8-s + (−6.48 − 3.74i)9-s + (−0.600 − 0.862i)10-s + 3.20i·11-s + (5.56 − 1.49i)12-s + (0.708 − 0.409i)13-s + (−1.24 + 1.24i)14-s + (4.66 − 5.53i)15-s + (−1.36 + 2.35i)16-s + (−0.330 + 0.571i)17-s + ⋯
L(s)  = 1  + (0.287 + 0.166i)2-s + (−0.483 + 1.80i)3-s + (−0.444 − 0.770i)4-s + (−0.905 − 0.425i)5-s + (−0.439 + 0.439i)6-s + (−0.365 + 1.36i)7-s − 0.628i·8-s + (−2.16 − 1.24i)9-s + (−0.189 − 0.272i)10-s + 0.967i·11-s + (1.60 − 0.430i)12-s + (0.196 − 0.113i)13-s + (−0.332 + 0.332i)14-s + (1.20 − 1.42i)15-s + (−0.340 + 0.589i)16-s + (−0.0800 + 0.138i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.986 - 0.161i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (162, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ -0.986 - 0.161i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0469496 + 0.578856i\)
\(L(\frac12)\) \(\approx\) \(0.0469496 + 0.578856i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.02 + 0.950i)T \)
37 \( 1 + (-2.61 - 5.49i)T \)
good2 \( 1 + (-0.407 - 0.235i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.838 - 3.12i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (0.968 - 3.61i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 - 3.20iT - 11T^{2} \)
13 \( 1 + (-0.708 + 0.409i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.330 - 0.571i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.437 + 1.63i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 4.26iT - 23T^{2} \)
29 \( 1 + (0.937 - 0.937i)T - 29iT^{2} \)
31 \( 1 + (-3.86 - 3.86i)T + 31iT^{2} \)
41 \( 1 + (-2.21 + 1.28i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 - 9.49iT - 43T^{2} \)
47 \( 1 + (6.02 + 6.02i)T + 47iT^{2} \)
53 \( 1 + (2.23 + 8.32i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.51 - 0.672i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.00 + 14.9i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (5.70 + 1.52i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.81 + 6.61i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.41 - 7.41i)T + 73iT^{2} \)
79 \( 1 + (3.47 - 12.9i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (2.60 + 9.71i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-1.41 - 5.28i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 - 3.89T + 97T^{2} \)
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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

−12.91923598811126480106174055262, −11.89669750546324669011773282238, −11.10667953568829052706302440322, −9.881905557999711067030587491344, −9.375578657493865114002919598196, −8.466369779485899593750026960046, −6.34232793992577587312821307464, −5.19521597068918352122054183701, −4.69249383725330819764187344522, −3.44258490738459653369143979779, 0.50952606001888710306238335865, 2.92117327213691450356035302788, 4.19893786758637728260630548519, 6.07190309755651776951351998785, 7.15848480973202164408758264753, 7.73620032639971085594691181547, 8.587494336693136793344758405913, 10.71137652162783750468808278764, 11.46730865200144099599584167011, 12.23900655755188514150924372003

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