Properties

Label 2-370-185.117-c1-0-9
Degree $2$
Conductor $370$
Sign $0.584 + 0.811i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
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  − 2-s + (0.0477 + 0.0477i)3-s + 4-s + (0.531 + 2.17i)5-s + (−0.0477 − 0.0477i)6-s + (−2.77 − 2.77i)7-s − 8-s − 2.99i·9-s + (−0.531 − 2.17i)10-s − 4.24i·11-s + (0.0477 + 0.0477i)12-s + 3.32·13-s + (2.77 + 2.77i)14-s + (−0.0783 + 0.129i)15-s + 16-s + 3.64i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.0275 + 0.0275i)3-s + 0.5·4-s + (0.237 + 0.971i)5-s + (−0.0194 − 0.0194i)6-s + (−1.05 − 1.05i)7-s − 0.353·8-s − 0.998i·9-s + (−0.167 − 0.686i)10-s − 1.27i·11-s + (0.0137 + 0.0137i)12-s + 0.920·13-s + (0.742 + 0.742i)14-s + (−0.0202 + 0.0333i)15-s + 0.250·16-s + 0.885i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.584 + 0.811i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.584 + 0.811i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.794232 - 0.406764i\)
\(L(\frac12)\) \(\approx\) \(0.794232 - 0.406764i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + (-0.531 - 2.17i)T \)
37 \( 1 + (3.91 + 4.65i)T \)
good3 \( 1 + (-0.0477 - 0.0477i)T + 3iT^{2} \)
7 \( 1 + (2.77 + 2.77i)T + 7iT^{2} \)
11 \( 1 + 4.24iT - 11T^{2} \)
13 \( 1 - 3.32T + 13T^{2} \)
17 \( 1 - 3.64iT - 17T^{2} \)
19 \( 1 + (-4.65 + 4.65i)T - 19iT^{2} \)
23 \( 1 - 3.99T + 23T^{2} \)
29 \( 1 + (-1.30 - 1.30i)T + 29iT^{2} \)
31 \( 1 + (-3.96 + 3.96i)T - 31iT^{2} \)
41 \( 1 + 2.70iT - 41T^{2} \)
43 \( 1 + 6.95T + 43T^{2} \)
47 \( 1 + (-1.34 - 1.34i)T + 47iT^{2} \)
53 \( 1 + (-5.37 + 5.37i)T - 53iT^{2} \)
59 \( 1 + (8.04 - 8.04i)T - 59iT^{2} \)
61 \( 1 + (-1.55 + 1.55i)T - 61iT^{2} \)
67 \( 1 + (4.34 - 4.34i)T - 67iT^{2} \)
71 \( 1 - 5.54T + 71T^{2} \)
73 \( 1 + (11.1 + 11.1i)T + 73iT^{2} \)
79 \( 1 + (3.17 - 3.17i)T - 79iT^{2} \)
83 \( 1 + (-7.08 + 7.08i)T - 83iT^{2} \)
89 \( 1 + (-7.75 - 7.75i)T + 89iT^{2} \)
97 \( 1 - 10.0iT - 97T^{2} \)
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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

−10.95272542244019722965714270088, −10.40304062800795779133709445054, −9.462681459138882007170038200365, −8.667096040637507036754364859263, −7.33172259418814759942422171000, −6.58196742422332003813551838770, −5.94753663994680924440991452181, −3.64427093801438612117726712609, −3.11749532968221996264122556597, −0.809098301223123814267765838036, 1.61756691077382866499540149540, 2.99085539826087008183437664917, 4.83400057248955721584412583751, 5.73504106887302150491699556892, 6.89703216075058402582738258957, 8.027018237312037712562042028310, 8.865747173053121096442613476270, 9.645758694597556779873518071531, 10.24888175187915735467714438810, 11.68969924664066353753072902590

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