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Properties

Label 2-370-185.103-c1-0-7
Degree $2$
Conductor $370$
Sign $-0.903 - 0.428i$
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  + (0.5 + 0.866i)2-s + (0.758 + 2.83i)3-s + (−0.499 + 0.866i)4-s + (0.917 + 2.03i)5-s + (−2.07 + 2.07i)6-s + (−0.490 − 1.83i)7-s − 0.999·8-s + (−4.84 + 2.79i)9-s + (−1.30 + 1.81i)10-s − 0.0963i·11-s + (−2.83 − 0.758i)12-s + (1.59 − 2.75i)13-s + (1.34 − 1.34i)14-s + (−5.07 + 4.14i)15-s + (−0.5 − 0.866i)16-s + (3.73 − 2.15i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.438 + 1.63i)3-s + (−0.249 + 0.433i)4-s + (0.410 + 0.911i)5-s + (−0.846 + 0.846i)6-s + (−0.185 − 0.692i)7-s − 0.353·8-s + (−1.61 + 0.932i)9-s + (−0.413 + 0.573i)10-s − 0.0290i·11-s + (−0.817 − 0.219i)12-s + (0.441 − 0.764i)13-s + (0.358 − 0.358i)14-s + (−1.31 + 1.07i)15-s + (−0.125 − 0.216i)16-s + (0.906 − 0.523i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 - 0.428i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.903 - 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.903 - 0.428i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.903 - 0.428i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.406320 + 1.80589i\)
\(L(\frac12)\) \(\approx\) \(0.406320 + 1.80589i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.917 - 2.03i)T \)
37 \( 1 + (5.72 + 2.05i)T \)
good3 \( 1 + (-0.758 - 2.83i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.490 + 1.83i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + 0.0963iT - 11T^{2} \)
13 \( 1 + (-1.59 + 2.75i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.73 + 2.15i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.71 + 1.26i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 1.96T + 23T^{2} \)
29 \( 1 + (3.50 - 3.50i)T - 29iT^{2} \)
31 \( 1 + (3.04 + 3.04i)T + 31iT^{2} \)
41 \( 1 + (-7.68 - 4.43i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + 7.57T + 43T^{2} \)
47 \( 1 + (9.56 - 9.56i)T - 47iT^{2} \)
53 \( 1 + (0.388 - 1.44i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.98 + 11.1i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-9.90 + 2.65i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (6.88 - 1.84i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.958 - 1.66i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.62 + 4.62i)T - 73iT^{2} \)
79 \( 1 + (2.89 - 0.775i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-3.32 + 12.4i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-4.69 - 1.25i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 - 17.6iT - 97T^{2} \)
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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

−11.41281344520688988890791789817, −10.71037284764205625680334327703, −9.850569212574383979128275387175, −9.314347672930253290701587082564, −8.004174093885852776326778215426, −7.09542788149141802252934713495, −5.75549169064282964508279155073, −4.95327222481244353272047857875, −3.57050936173512782348863055662, −3.13298486655159994255646131664, 1.24309210907284898626155213856, 2.14293626902417542608089902027, 3.53303574467963588425094559025, 5.31194307648342477875989713907, 6.05910322922075029269085134556, 7.21212918310928966767547015999, 8.368741733978146588616650062733, 8.989078764551651665062326914224, 9.968335795322151899410382494809, 11.56759631507770612517027457490

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