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Properties

Label 2-370-185.68-c1-0-7
Degree $2$
Conductor $370$
Sign $0.393 - 0.919i$
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 + (−1.70 + 1.70i)3-s + 4-s + (1.73 + 1.41i)5-s + (−1.70 + 1.70i)6-s + (2.82 − 2.82i)7-s + 8-s − 2.79i·9-s + (1.73 + 1.41i)10-s + 2.94i·11-s + (−1.70 + 1.70i)12-s − 0.738·13-s + (2.82 − 2.82i)14-s + (−5.35 + 0.542i)15-s + 16-s + 6.61i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.982 + 0.982i)3-s + 0.5·4-s + (0.774 + 0.632i)5-s + (−0.694 + 0.694i)6-s + (1.06 − 1.06i)7-s + 0.353·8-s − 0.930i·9-s + (0.547 + 0.447i)10-s + 0.887i·11-s + (−0.491 + 0.491i)12-s − 0.204·13-s + (0.755 − 0.755i)14-s + (−1.38 + 0.140i)15-s + 0.250·16-s + 1.60i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.393 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.393 - 0.919i$
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: \((2,\ 370,\ (\ :1/2),\ 0.393 - 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52162 + 1.00385i\)
\(L(\frac12)\) \(\approx\) \(1.52162 + 1.00385i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - 1.41i)T \)
37 \( 1 + (6.07 + 0.209i)T \)
good3 \( 1 + (1.70 - 1.70i)T - 3iT^{2} \)
7 \( 1 + (-2.82 + 2.82i)T - 7iT^{2} \)
11 \( 1 - 2.94iT - 11T^{2} \)
13 \( 1 + 0.738T + 13T^{2} \)
17 \( 1 - 6.61iT - 17T^{2} \)
19 \( 1 + (5.03 + 5.03i)T + 19iT^{2} \)
23 \( 1 - 6.39T + 23T^{2} \)
29 \( 1 + (2.49 - 2.49i)T - 29iT^{2} \)
31 \( 1 + (1.48 + 1.48i)T + 31iT^{2} \)
41 \( 1 + 7.39iT - 41T^{2} \)
43 \( 1 - 6.98T + 43T^{2} \)
47 \( 1 + (-7.56 + 7.56i)T - 47iT^{2} \)
53 \( 1 + (-1.94 - 1.94i)T + 53iT^{2} \)
59 \( 1 + (8.34 + 8.34i)T + 59iT^{2} \)
61 \( 1 + (2.31 + 2.31i)T + 61iT^{2} \)
67 \( 1 + (-2.10 - 2.10i)T + 67iT^{2} \)
71 \( 1 + 2.52T + 71T^{2} \)
73 \( 1 + (-7.18 + 7.18i)T - 73iT^{2} \)
79 \( 1 + (5.69 + 5.69i)T + 79iT^{2} \)
83 \( 1 + (2.72 + 2.72i)T + 83iT^{2} \)
89 \( 1 + (4.25 - 4.25i)T - 89iT^{2} \)
97 \( 1 + 3.40iT - 97T^{2} \)
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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

−11.15737695047250237433619460704, −10.69210833163592777316423696810, −10.37886473533915930696255731245, −9.036990982460263624484892536554, −7.41872562294935912370962821180, −6.62015377020971433413944028712, −5.46933099833907520107805105239, −4.69241672358564267769035736834, −3.87890341900414032635764239625, −1.96598045088653416350325182370, 1.31393646799969485079497721304, 2.53782800269022233970606775193, 4.71711461154622089720219883329, 5.56004858696407903666631020742, 6.00408968524519918664570533593, 7.19054455902537695605305009701, 8.341186017179764953201852765588, 9.269850028927769129611484316697, 10.82294679249585216200749794675, 11.48268391415307120845377061304

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