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Properties

Label 2-185-185.84-c1-0-15
Degree $2$
Conductor $185$
Sign $0.936 + 0.349i$
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  + (0.843 + 0.486i)2-s + (1.64 − 0.952i)3-s + (−0.526 − 0.911i)4-s + (2.05 − 0.885i)5-s + 1.85·6-s + (−3.89 + 2.24i)7-s − 2.97i·8-s + (0.314 − 0.544i)9-s + (2.16 + 0.253i)10-s + 3.42·11-s + (−1.73 − 1.00i)12-s + (−4.09 + 2.36i)13-s − 4.37·14-s + (2.54 − 3.41i)15-s + (0.394 − 0.683i)16-s + (0.520 + 0.300i)17-s + ⋯
L(s)  = 1  + (0.596 + 0.344i)2-s + (0.952 − 0.549i)3-s + (−0.263 − 0.455i)4-s + (0.918 − 0.395i)5-s + 0.757·6-s + (−1.47 + 0.849i)7-s − 1.05i·8-s + (0.104 − 0.181i)9-s + (0.683 + 0.0800i)10-s + 1.03·11-s + (−0.501 − 0.289i)12-s + (−1.13 + 0.656i)13-s − 1.16·14-s + (0.656 − 0.882i)15-s + (0.0986 − 0.170i)16-s + (0.126 + 0.0728i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.936 + 0.349i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (84, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ 0.936 + 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86230 - 0.336021i\)
\(L(\frac12)\) \(\approx\) \(1.86230 - 0.336021i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.05 + 0.885i)T \)
37 \( 1 + (3.72 + 4.80i)T \)
good2 \( 1 + (-0.843 - 0.486i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.64 + 0.952i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (3.89 - 2.24i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 3.42T + 11T^{2} \)
13 \( 1 + (4.09 - 2.36i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.520 - 0.300i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.78 - 4.81i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.727iT - 23T^{2} \)
29 \( 1 - 5.69T + 29T^{2} \)
31 \( 1 + 6.49T + 31T^{2} \)
41 \( 1 + (2.32 + 4.02i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 5.47iT - 43T^{2} \)
47 \( 1 + 2.57iT - 47T^{2} \)
53 \( 1 + (6.49 + 3.74i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.641 - 1.11i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.24 + 7.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.00 - 2.88i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.71 + 2.96i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.39iT - 73T^{2} \)
79 \( 1 + (-8.30 - 14.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.57 - 5.52i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.80 + 8.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.09iT - 97T^{2} \)
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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

−12.61569082664088502741524319033, −12.32507018092526289195400283951, −10.09404861414166054205148460127, −9.388771087020261199224769719691, −8.844995075419928842856588891180, −7.06835635769986065125970821282, −6.21225264825494480363100806457, −5.23001033840307267261244953469, −3.50666839920793630899000382702, −1.98876352629010638086213970098, 2.88070388985986765193203377037, 3.37322299181976504296012484734, 4.73330812915886364470283557854, 6.36996671794580505522644751826, 7.48612696603262833259948297559, 9.089489995712426679739695907132, 9.555625454938918064995517065257, 10.45889060766739575559999391436, 11.93001179771138109880840927334, 12.95833565357231075515116174390

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