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Properties

Label 2-370-185.142-c1-0-14
Degree $2$
Conductor $370$
Sign $-0.571 + 0.820i$
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  − i·2-s + (−0.0477 − 0.0477i)3-s − 4-s + (2.17 − 0.531i)5-s + (−0.0477 + 0.0477i)6-s + (−2.77 − 2.77i)7-s + i·8-s − 2.99i·9-s + (−0.531 − 2.17i)10-s + 4.24i·11-s + (0.0477 + 0.0477i)12-s − 3.32i·13-s + (−2.77 + 2.77i)14-s + (−0.129 − 0.0783i)15-s + 16-s + 3.64·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.0275 − 0.0275i)3-s − 0.5·4-s + (0.971 − 0.237i)5-s + (−0.0194 + 0.0194i)6-s + (−1.05 − 1.05i)7-s + 0.353i·8-s − 0.998i·9-s + (−0.167 − 0.686i)10-s + 1.27i·11-s + (0.0137 + 0.0137i)12-s − 0.920i·13-s + (−0.742 + 0.742i)14-s + (−0.0333 − 0.0202i)15-s + 0.250·16-s + 0.885·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 + 0.820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.571 + 0.820i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.571 + 0.820i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.568940 - 1.08935i\)
\(L(\frac12)\) \(\approx\) \(0.568940 - 1.08935i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (-2.17 + 0.531i)T \)
37 \( 1 + (4.65 + 3.91i)T \)
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 \( 1 + 3.32iT - 13T^{2} \)
17 \( 1 - 3.64T + 17T^{2} \)
19 \( 1 + (4.65 + 4.65i)T + 19iT^{2} \)
23 \( 1 + 3.99iT - 23T^{2} \)
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 \( 1 - 6.95iT - 43T^{2} \)
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 \( 1 - 10.0T + 97T^{2} \)
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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

−10.79789231196650359103417012330, −10.03568890739906929804515617195, −9.671457015759420401322977901588, −8.608925269304686942373078416672, −7.08335097552315701816506367805, −6.35580736741268656210116807974, −5.00105059439879607412715306216, −3.82158945151758078693792664252, −2.61117089670679367428637336462, −0.859022198824899713090228786327, 2.18654795156572766397752280003, 3.57843099190239296933315067847, 5.36139058433293875215452829861, 5.91657866263887261957682221002, 6.69136005933664217358272964260, 8.086473454193845677899967866586, 8.914066901663222643033549068203, 9.754611799721319879503867984704, 10.55648539612453107233330971531, 11.79155481834740815018459678951

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