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Properties

Label 2-370-185.162-c1-0-7
Degree $2$
Conductor $370$
Sign $0.866 - 0.499i$
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  + (−0.866 − 0.5i)2-s + (−0.392 + 1.46i)3-s + (0.499 + 0.866i)4-s + (2.15 − 0.582i)5-s + (1.07 − 1.07i)6-s + (−0.155 + 0.578i)7-s − 0.999i·8-s + (0.603 + 0.348i)9-s + (−2.16 − 0.575i)10-s − 2.03i·11-s + (−1.46 + 0.392i)12-s + (3.91 − 2.26i)13-s + (0.423 − 0.423i)14-s + (0.00519 + 3.39i)15-s + (−0.5 + 0.866i)16-s + (−3.22 + 5.58i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.226 + 0.846i)3-s + (0.249 + 0.433i)4-s + (0.965 − 0.260i)5-s + (0.438 − 0.438i)6-s + (−0.0586 + 0.218i)7-s − 0.353i·8-s + (0.201 + 0.116i)9-s + (−0.683 − 0.181i)10-s − 0.613i·11-s + (−0.423 + 0.113i)12-s + (1.08 − 0.626i)13-s + (0.113 − 0.113i)14-s + (0.00134 + 0.876i)15-s + (−0.125 + 0.216i)16-s + (−0.782 + 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.866 - 0.499i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.866 - 0.499i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12072 + 0.299925i\)
\(L(\frac12)\) \(\approx\) \(1.12072 + 0.299925i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-2.15 + 0.582i)T \)
37 \( 1 + (0.586 - 6.05i)T \)
good3 \( 1 + (0.392 - 1.46i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (0.155 - 0.578i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + 2.03iT - 11T^{2} \)
13 \( 1 + (-3.91 + 2.26i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.22 - 5.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.311 + 1.16i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 4.39iT - 23T^{2} \)
29 \( 1 + (-5.43 + 5.43i)T - 29iT^{2} \)
31 \( 1 + (-5.48 - 5.48i)T + 31iT^{2} \)
41 \( 1 + (1.10 - 0.637i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + 3.83iT - 43T^{2} \)
47 \( 1 + (3.13 + 3.13i)T + 47iT^{2} \)
53 \( 1 + (-0.0992 - 0.370i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (10.2 - 2.74i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.46 + 9.19i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (0.762 + 0.204i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (8.36 + 14.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.81 + 3.81i)T + 73iT^{2} \)
79 \( 1 + (3.16 - 11.8i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (-0.261 - 0.974i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (2.11 + 7.88i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + 15.7T + 97T^{2} \)
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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

−11.03980624239536191493509327200, −10.48994812804442072418717643821, −9.809684698718677408322655539572, −8.824738414151242583639113290228, −8.196118789434269480066457389312, −6.52235634787424048203468229067, −5.69830625953032433156244366103, −4.45637739941817032854969133438, −3.15497929333184623340605459469, −1.51951803219766543862604745407, 1.20655725215176595612969497953, 2.46414236861265247603398714849, 4.52251403219934752794624015610, 5.96731499121195121756077058981, 6.71266378779063793661956644507, 7.25286968081813085529163602062, 8.592793176099152788864117259841, 9.450018023096190922615914325846, 10.27024660714032406195604013966, 11.22674815534194572226889783365

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