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Properties

Label 2-370-185.82-c1-0-3
Degree $2$
Conductor $370$
Sign $0.934 - 0.355i$
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 + (−1.46 + 0.392i)3-s + (−0.499 + 0.866i)4-s + (−1.97 − 1.04i)5-s + (1.07 + 1.07i)6-s + (2.06 − 0.552i)7-s + 0.999·8-s + (−0.603 + 0.348i)9-s + (0.0876 + 2.23i)10-s + 1.24i·11-s + (0.392 − 1.46i)12-s + (−2.08 + 3.60i)13-s + (−1.50 − 1.50i)14-s + (3.30 + 0.749i)15-s + (−0.5 − 0.866i)16-s + (4.89 − 2.82i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.846 + 0.226i)3-s + (−0.249 + 0.433i)4-s + (−0.884 − 0.465i)5-s + (0.438 + 0.438i)6-s + (0.778 − 0.208i)7-s + 0.353·8-s + (−0.201 + 0.116i)9-s + (0.0277 + 0.706i)10-s + 0.374i·11-s + (0.113 − 0.423i)12-s + (−0.577 + 1.00i)13-s + (−0.403 − 0.403i)14-s + (0.854 + 0.193i)15-s + (−0.125 − 0.216i)16-s + (1.18 − 0.685i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.934 - 0.355i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.934 - 0.355i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.647602 + 0.118879i\)
\(L(\frac12)\) \(\approx\) \(0.647602 + 0.118879i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.97 + 1.04i)T \)
37 \( 1 + (-6.05 + 0.586i)T \)
good3 \( 1 + (1.46 - 0.392i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-2.06 + 0.552i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 - 1.24iT - 11T^{2} \)
13 \( 1 + (2.08 - 3.60i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-4.89 + 2.82i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.647 - 2.41i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 7.58T + 23T^{2} \)
29 \( 1 + (-7.29 - 7.29i)T + 29iT^{2} \)
31 \( 1 + (6.72 - 6.72i)T - 31iT^{2} \)
41 \( 1 + (-4.66 - 2.69i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + 6.61T + 43T^{2} \)
47 \( 1 + (3.36 + 3.36i)T + 47iT^{2} \)
53 \( 1 + (6.96 + 1.86i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (11.7 + 3.14i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.279 - 1.04i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.54 - 9.47i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-4.90 + 8.49i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-10.7 - 10.7i)T + 73iT^{2} \)
79 \( 1 + (0.454 + 1.69i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-0.110 - 0.0295i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (-1.84 + 6.87i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 - 0.402iT - 97T^{2} \)
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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

−11.34433923476486030356336909844, −10.87601158244736431738678140866, −9.735889426474665233189445918611, −8.749963513852808453636975411192, −7.80302391819950518974124488922, −6.91304527389067882982545507120, −5.04692759041974757314793046927, −4.76180149129008326846760960872, −3.22125604892601403486039587841, −1.24432035568588631897255760276, 0.67380265247402976577320956121, 3.08200398614686247612406707337, 4.73066514431192125904790762790, 5.63967720022507903724789992861, 6.56725266885457161474545118226, 7.74473263639761137922936020628, 8.134604227743959516151773215682, 9.452746684926963261557635836706, 10.71859419791131472329216858613, 11.20853844062479158621862453501

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