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Properties

Label 2-370-185.103-c1-0-4
Degree $2$
Conductor $370$
Sign $-0.0446 - 0.999i$
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 + (0.608 + 2.26i)3-s + (−0.499 + 0.866i)4-s + (0.642 + 2.14i)5-s + (1.66 − 1.66i)6-s + (0.265 + 0.991i)7-s + 0.999·8-s + (−2.18 + 1.26i)9-s + (1.53 − 1.62i)10-s − 1.09i·11-s + (−2.26 − 0.608i)12-s + (−0.227 + 0.393i)13-s + (0.725 − 0.725i)14-s + (−4.47 + 2.76i)15-s + (−0.5 − 0.866i)16-s + (−2.50 + 1.44i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.351 + 1.31i)3-s + (−0.249 + 0.433i)4-s + (0.287 + 0.957i)5-s + (0.678 − 0.678i)6-s + (0.100 + 0.374i)7-s + 0.353·8-s + (−0.728 + 0.420i)9-s + (0.484 − 0.514i)10-s − 0.329i·11-s + (−0.655 − 0.175i)12-s + (−0.0630 + 0.109i)13-s + (0.193 − 0.193i)14-s + (−1.15 + 0.712i)15-s + (−0.125 − 0.216i)16-s + (−0.607 + 0.350i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0446 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0446 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.0446 - 0.999i$
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: \((2,\ 370,\ (\ :1/2),\ -0.0446 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.841364 + 0.879782i\)
\(L(\frac12)\) \(\approx\) \(0.841364 + 0.879782i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.642 - 2.14i)T \)
37 \( 1 + (-1.64 - 5.85i)T \)
good3 \( 1 + (-0.608 - 2.26i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.265 - 0.991i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + 1.09iT - 11T^{2} \)
13 \( 1 + (0.227 - 0.393i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.50 - 1.44i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.134 - 0.0359i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 0.390T + 23T^{2} \)
29 \( 1 + (-1.28 + 1.28i)T - 29iT^{2} \)
31 \( 1 + (0.795 + 0.795i)T + 31iT^{2} \)
41 \( 1 + (6.11 + 3.52i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 8.52T + 43T^{2} \)
47 \( 1 + (-6.69 + 6.69i)T - 47iT^{2} \)
53 \( 1 + (2.03 - 7.58i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.934 - 3.48i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.90 + 1.31i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-12.3 + 3.30i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.63 + 6.29i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.40 + 1.40i)T - 73iT^{2} \)
79 \( 1 + (4.98 - 1.33i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-1.08 + 4.04i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (13.3 + 3.57i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 - 11.8iT - 97T^{2} \)
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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

−11.22979654556435346369231963847, −10.61323692049724676757902577154, −9.921106195630512127175286850730, −9.134826983813261784599601786889, −8.315217596798200430475538548417, −6.96527709032730779405422757591, −5.66967582712535823143111681436, −4.34303037416095151091171846296, −3.38934566555039601788548844556, −2.32450652185917877178182145033, 0.935008465119900327403950371748, 2.21960460649217296471976422641, 4.34674750200951040295678536508, 5.53129453791748466896112479907, 6.64736064723803546355471581050, 7.43216870351667891979366198575, 8.233610220675610015419308494402, 9.018779315691784624524802528259, 9.965056113966507935354094944187, 11.23477062547265043957889053563

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