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Properties

Label 2-185-185.103-c1-0-6
Degree $2$
Conductor $185$
Sign $0.909 - 0.415i$
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  + (0.647 + 1.12i)2-s + (−0.366 − 1.36i)3-s + (0.161 − 0.279i)4-s + (0.213 + 2.22i)5-s + (1.29 − 1.29i)6-s + (0.0333 + 0.124i)7-s + 3.00·8-s + (0.861 − 0.497i)9-s + (−2.35 + 1.68i)10-s + 2.33i·11-s + (−0.440 − 0.118i)12-s + (2.86 − 4.96i)13-s + (−0.117 + 0.117i)14-s + (2.96 − 1.10i)15-s + (1.62 + 2.81i)16-s + (−4.88 + 2.81i)17-s + ⋯
L(s)  = 1  + (0.457 + 0.793i)2-s + (−0.211 − 0.789i)3-s + (0.0805 − 0.139i)4-s + (0.0953 + 0.995i)5-s + (0.529 − 0.529i)6-s + (0.0125 + 0.0469i)7-s + 1.06·8-s + (0.287 − 0.165i)9-s + (−0.745 + 0.531i)10-s + 0.704i·11-s + (−0.127 − 0.0340i)12-s + (0.795 − 1.37i)13-s + (−0.0315 + 0.0315i)14-s + (0.765 − 0.285i)15-s + (0.406 + 0.704i)16-s + (−1.18 + 0.683i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.909 - 0.415i$
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: \((2,\ 185,\ (\ :1/2),\ 0.909 - 0.415i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50819 + 0.327730i\)
\(L(\frac12)\) \(\approx\) \(1.50819 + 0.327730i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.213 - 2.22i)T \)
37 \( 1 + (-1.97 - 5.75i)T \)
good2 \( 1 + (-0.647 - 1.12i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.366 + 1.36i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.0333 - 0.124i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 - 2.33iT - 11T^{2} \)
13 \( 1 + (-2.86 + 4.96i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.88 - 2.81i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.92 + 0.516i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 9.37T + 23T^{2} \)
29 \( 1 + (4.17 - 4.17i)T - 29iT^{2} \)
31 \( 1 + (2.28 + 2.28i)T + 31iT^{2} \)
41 \( 1 + (-5.79 - 3.34i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + 9.62T + 43T^{2} \)
47 \( 1 + (-7.16 + 7.16i)T - 47iT^{2} \)
53 \( 1 + (-0.612 + 2.28i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.04 + 3.88i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.21 + 0.326i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (6.26 - 1.67i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.22 - 5.57i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.20 - 2.20i)T - 73iT^{2} \)
79 \( 1 + (-3.38 + 0.906i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-1.44 + 5.39i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (3.70 + 0.992i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 - 3.93iT - 97T^{2} \)
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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

−13.07478788480144069593519918123, −11.77916781494807049347571634816, −10.67760193045856282689027250144, −9.959272563315679332145329340416, −8.078290452843678363519765362416, −7.23227386900222280649156582086, −6.41473284758746470946030829658, −5.67126406315683137467916166954, −3.95431761965065674085441713777, −1.93495653028011217896089890772, 1.92219325224104603844176171827, 3.95890753261232647499249438674, 4.42176374693340361211134894725, 5.83764944654824884889940205163, 7.49272658403424665692324598077, 8.800862563979873134090795629596, 9.667553248491369884964147131907, 10.87723862099316603341098366158, 11.51810312095470414828948110744, 12.38133144214664639590511569591

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