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Properties

Label 2-370-185.68-c1-0-4
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: $\chi_{370} (253, \cdot )$
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

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

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