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Properties

Label 2-185-185.43-c1-0-7
Degree $2$
Conductor $185$
Sign $0.898 - 0.438i$
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  + 0.311i·2-s + (0.762 − 0.762i)3-s + 1.90·4-s + (0.311 + 2.21i)5-s + (0.237 + 0.237i)6-s + (−1.76 + 1.76i)7-s + 1.21i·8-s + 1.83i·9-s + (−0.688 + 0.0967i)10-s − 4.28i·11-s + (1.45 − 1.45i)12-s − 4.83i·13-s + (−0.548 − 0.548i)14-s + (1.92 + 1.45i)15-s + 3.42·16-s − 3.59·17-s + ⋯
L(s)  = 1  + 0.219i·2-s + (0.440 − 0.440i)3-s + 0.951·4-s + (0.139 + 0.990i)5-s + (0.0968 + 0.0968i)6-s + (−0.666 + 0.666i)7-s + 0.429i·8-s + 0.612i·9-s + (−0.217 + 0.0306i)10-s − 1.29i·11-s + (0.419 − 0.419i)12-s − 1.34i·13-s + (−0.146 − 0.146i)14-s + (0.497 + 0.374i)15-s + 0.857·16-s − 0.871·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.898 - 0.438i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ 0.898 - 0.438i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48217 + 0.342340i\)
\(L(\frac12)\) \(\approx\) \(1.48217 + 0.342340i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.311 - 2.21i)T \)
37 \( 1 + (1 + 6i)T \)
good2 \( 1 - 0.311iT - 2T^{2} \)
3 \( 1 + (-0.762 + 0.762i)T - 3iT^{2} \)
7 \( 1 + (1.76 - 1.76i)T - 7iT^{2} \)
11 \( 1 + 4.28iT - 11T^{2} \)
13 \( 1 + 4.83iT - 13T^{2} \)
17 \( 1 + 3.59T + 17T^{2} \)
19 \( 1 + (-1.23 + 1.23i)T - 19iT^{2} \)
23 \( 1 - 0.280iT - 23T^{2} \)
29 \( 1 + (5.42 + 5.42i)T + 29iT^{2} \)
31 \( 1 + (-5.56 + 5.56i)T - 31iT^{2} \)
41 \( 1 - 12.0iT - 41T^{2} \)
43 \( 1 + 3.65iT - 43T^{2} \)
47 \( 1 + (6.88 - 6.88i)T - 47iT^{2} \)
53 \( 1 + (2.31 + 2.31i)T + 53iT^{2} \)
59 \( 1 + (1.56 - 1.56i)T - 59iT^{2} \)
61 \( 1 + (-2.09 + 2.09i)T - 61iT^{2} \)
67 \( 1 + (-0.400 - 0.400i)T + 67iT^{2} \)
71 \( 1 - 5.18T + 71T^{2} \)
73 \( 1 + (10.7 - 10.7i)T - 73iT^{2} \)
79 \( 1 + (2.92 - 2.92i)T - 79iT^{2} \)
83 \( 1 + (-4.07 - 4.07i)T + 83iT^{2} \)
89 \( 1 + (-5.64 - 5.64i)T + 89iT^{2} \)
97 \( 1 - 1.52T + 97T^{2} \)
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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

−12.89549905884231034171133204524, −11.42790779951279270783018533439, −10.94369627645752629866646609423, −9.791421391039267547669753567338, −8.293297818191121361411240720032, −7.57084571042269068164809349595, −6.37168238125603351793417893014, −5.71374780943575609470727471395, −3.13093798439865244708666706878, −2.45540837306857380680161182328, 1.79950073499897033955919851265, 3.58550168408044011024846684626, 4.66906855738295981465207300110, 6.48083200342084485608789702288, 7.19637763194255434910501728215, 8.785779206768382134824671898847, 9.614841197452699973215058490065, 10.35031211284227570259857170651, 11.76945186078309276329114646405, 12.38215887447012469932119673958

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