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Properties

Label 2-185-185.23-c1-0-10
Degree $2$
Conductor $185$
Sign $0.942 + 0.335i$
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  + (−0.305 − 0.176i)2-s + (2.52 + 0.676i)3-s + (−0.937 − 1.62i)4-s + (2.22 − 0.204i)5-s + (−0.652 − 0.652i)6-s + (−1.50 − 0.404i)7-s + 1.36i·8-s + (3.31 + 1.91i)9-s + (−0.716 − 0.330i)10-s − 2.57i·11-s + (−1.26 − 4.73i)12-s + (−4.41 + 2.54i)13-s + (0.389 + 0.389i)14-s + (5.76 + 0.989i)15-s + (−1.63 + 2.83i)16-s + (−0.342 + 0.593i)17-s + ⋯
L(s)  = 1  + (−0.216 − 0.124i)2-s + (1.45 + 0.390i)3-s + (−0.468 − 0.812i)4-s + (0.995 − 0.0915i)5-s + (−0.266 − 0.266i)6-s + (−0.570 − 0.152i)7-s + 0.483i·8-s + (1.10 + 0.638i)9-s + (−0.226 − 0.104i)10-s − 0.775i·11-s + (−0.366 − 1.36i)12-s + (−1.22 + 0.706i)13-s + (0.104 + 0.104i)14-s + (1.48 + 0.255i)15-s + (−0.408 + 0.707i)16-s + (−0.0830 + 0.143i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.942 + 0.335i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ 0.942 + 0.335i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52414 - 0.263202i\)
\(L(\frac12)\) \(\approx\) \(1.52414 - 0.263202i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.22 + 0.204i)T \)
37 \( 1 + (-4.74 + 3.80i)T \)
good2 \( 1 + (0.305 + 0.176i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-2.52 - 0.676i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (1.50 + 0.404i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + 2.57iT - 11T^{2} \)
13 \( 1 + (4.41 - 2.54i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.342 - 0.593i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.59 - 1.23i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 5.83iT - 23T^{2} \)
29 \( 1 + (4.71 + 4.71i)T + 29iT^{2} \)
31 \( 1 + (1.40 - 1.40i)T - 31iT^{2} \)
41 \( 1 + (6.82 - 3.94i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 - 2.28iT - 43T^{2} \)
47 \( 1 + (7.99 - 7.99i)T - 47iT^{2} \)
53 \( 1 + (-4.37 + 1.17i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.43 + 9.07i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.49 + 0.667i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.100 + 0.376i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.0695 - 0.120i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.70 + 6.70i)T - 73iT^{2} \)
79 \( 1 + (-0.455 - 0.121i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (3.06 - 0.821i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-15.2 + 4.08i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 - 13.7T + 97T^{2} \)
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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

−13.06737945478271197123334621268, −11.34929883948415060386521835468, −9.843441159485964443706878697299, −9.696961104886147825580163539396, −8.957126103293963279024209915317, −7.70321717436705975496401297173, −6.14970203167096639725976956276, −4.92331207080048766702302856057, −3.35156204117087993513656439631, −1.93634073215538490933220753292, 2.37767052949276498156148617185, 3.28077887505769305922674400179, 5.00646112470128736338583050977, 6.89450415779111495602573596927, 7.59884879885618363136631311419, 8.728478248119460470265857280214, 9.480715764254436244144602146295, 10.09116246474969126451968714288, 12.17529774049293676703275371775, 12.95758183052952261072452005072

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