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Properties

Label 2-370-185.88-c1-0-15
Degree $2$
Conductor $370$
Sign $0.599 + 0.800i$
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 + (3.09 + 0.830i)3-s + (−0.499 − 0.866i)4-s + (−1.21 − 1.87i)5-s + (2.26 − 2.26i)6-s + (−1.35 − 0.362i)7-s − 0.999·8-s + (6.31 + 3.64i)9-s + (−2.23 + 0.117i)10-s − 4.96i·11-s + (−0.830 − 3.09i)12-s + (2.91 + 5.05i)13-s + (−0.990 + 0.990i)14-s + (−2.21 − 6.82i)15-s + (−0.5 + 0.866i)16-s + (1.69 + 0.981i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (1.78 + 0.479i)3-s + (−0.249 − 0.433i)4-s + (−0.544 − 0.838i)5-s + (0.925 − 0.925i)6-s + (−0.511 − 0.137i)7-s − 0.353·8-s + (2.10 + 1.21i)9-s + (−0.706 + 0.0372i)10-s − 1.49i·11-s + (−0.239 − 0.894i)12-s + (0.808 + 1.40i)13-s + (−0.264 + 0.264i)14-s + (−0.572 − 1.76i)15-s + (−0.125 + 0.216i)16-s + (0.412 + 0.237i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.599 + 0.800i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.599 + 0.800i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18143 - 1.09152i\)
\(L(\frac12)\) \(\approx\) \(2.18143 - 1.09152i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.21 + 1.87i)T \)
37 \( 1 + (4.16 - 4.43i)T \)
good3 \( 1 + (-3.09 - 0.830i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (1.35 + 0.362i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + 4.96iT - 11T^{2} \)
13 \( 1 + (-2.91 - 5.05i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.69 - 0.981i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.486 + 1.81i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 7.69T + 23T^{2} \)
29 \( 1 + (2.99 - 2.99i)T - 29iT^{2} \)
31 \( 1 + (-6.39 - 6.39i)T + 31iT^{2} \)
41 \( 1 + (1.83 - 1.06i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + 4.29T + 43T^{2} \)
47 \( 1 + (-1.20 + 1.20i)T - 47iT^{2} \)
53 \( 1 + (5.68 - 1.52i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.56 + 0.687i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.83 + 6.84i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-3.08 + 11.5i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.0977 - 0.169i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.35 + 1.35i)T - 73iT^{2} \)
79 \( 1 + (-0.103 + 0.387i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (-8.11 + 2.17i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (0.518 + 1.93i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 - 14.5iT - 97T^{2} \)
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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

−11.28423680681411968930682840194, −10.16653394597151048218947676128, −9.280199041381128836393258686355, −8.617608700656165202971243205147, −8.061370219687038151661789961381, −6.49009256712764598603298772547, −4.84104508359810422746426114597, −3.73818731667962230528768691033, −3.32625875403616618311332908061, −1.64632550145964761384363406004, 2.31095829141197295376015551601, 3.36910237112846389670832507731, 4.12239635454671264454861490946, 6.05459035159932163608935034235, 7.10435943966717974772508770316, 7.82149893712848471250981037680, 8.282989035408301038095436007403, 9.672992819163673075414167122281, 10.15799505845646713294482773764, 11.94559700506532261039695758503

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