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Properties

Label 2-370-185.43-c1-0-14
Degree $2$
Conductor $370$
Sign $0.668 + 0.744i$
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  + i·2-s + (1.23 − 1.23i)3-s − 4-s + (−1.84 − 1.26i)5-s + (1.23 + 1.23i)6-s + (1.28 − 1.28i)7-s − i·8-s − 0.0591i·9-s + (1.26 − 1.84i)10-s − 4.27i·11-s + (−1.23 + 1.23i)12-s − 1.50i·13-s + (1.28 + 1.28i)14-s + (−3.84 + 0.708i)15-s + 16-s + 2.12·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.714 − 0.714i)3-s − 0.5·4-s + (−0.823 − 0.567i)5-s + (0.504 + 0.504i)6-s + (0.486 − 0.486i)7-s − 0.353i·8-s − 0.0197i·9-s + (0.401 − 0.582i)10-s − 1.28i·11-s + (−0.357 + 0.357i)12-s − 0.418i·13-s + (0.343 + 0.343i)14-s + (−0.993 + 0.182i)15-s + 0.250·16-s + 0.514·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 + 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.668 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.668 + 0.744i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.668 + 0.744i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28306 - 0.572254i\)
\(L(\frac12)\) \(\approx\) \(1.28306 - 0.572254i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (1.84 + 1.26i)T \)
37 \( 1 + (2.28 - 5.63i)T \)
good3 \( 1 + (-1.23 + 1.23i)T - 3iT^{2} \)
7 \( 1 + (-1.28 + 1.28i)T - 7iT^{2} \)
11 \( 1 + 4.27iT - 11T^{2} \)
13 \( 1 + 1.50iT - 13T^{2} \)
17 \( 1 - 2.12T + 17T^{2} \)
19 \( 1 + (-0.381 + 0.381i)T - 19iT^{2} \)
23 \( 1 + 8.90iT - 23T^{2} \)
29 \( 1 + (0.279 + 0.279i)T + 29iT^{2} \)
31 \( 1 + (1.59 - 1.59i)T - 31iT^{2} \)
41 \( 1 - 0.270iT - 41T^{2} \)
43 \( 1 - 0.302iT - 43T^{2} \)
47 \( 1 + (3.17 - 3.17i)T - 47iT^{2} \)
53 \( 1 + (-9.71 - 9.71i)T + 53iT^{2} \)
59 \( 1 + (-4.06 + 4.06i)T - 59iT^{2} \)
61 \( 1 + (-4.47 + 4.47i)T - 61iT^{2} \)
67 \( 1 + (-6.17 - 6.17i)T + 67iT^{2} \)
71 \( 1 + 7.90T + 71T^{2} \)
73 \( 1 + (8.32 - 8.32i)T - 73iT^{2} \)
79 \( 1 + (-2.20 + 2.20i)T - 79iT^{2} \)
83 \( 1 + (-10.1 - 10.1i)T + 83iT^{2} \)
89 \( 1 + (5.24 + 5.24i)T + 89iT^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−11.31443543680421253778107582443, −10.38062148905224031196262295719, −8.808166912099767027716054113701, −8.320492072968178873293650095749, −7.72863802619521766549494678860, −6.78440096488479489001759360606, −5.44171712454659477997841502850, −4.33284961132573106607445440743, −3.04488463931831638400949357870, −0.962423269816020330762629472929, 2.08560810095328927114473618733, 3.41933805246751678198936054029, 4.13603693007502102603432265577, 5.30414617961227368343024652389, 7.05059428729905584482955709941, 7.963063503936552409877419475373, 8.988914262413312148997822341752, 9.732013943245971155525911980193, 10.50417781470717928859845312974, 11.73748835641017276011467035773

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