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Properties

Label 1-740-740.343-r0-0-0
Degree $1$
Conductor $740$
Sign $-0.993 - 0.116i$
Analytic cond. $3.43654$
Root an. cond. $3.43654$
Motivic weight $0$
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)3-s + (−0.866 − 0.5i)7-s + (0.5 + 0.866i)9-s − 11-s + (−0.866 − 0.5i)13-s + (−0.866 + 0.5i)17-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)21-s − i·23-s − i·27-s − 29-s − 31-s + (−0.866 − 0.5i)33-s + (−0.5 − 0.866i)39-s + (−0.5 + 0.866i)41-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (−0.866 − 0.5i)7-s + (0.5 + 0.866i)9-s − 11-s + (−0.866 − 0.5i)13-s + (−0.866 + 0.5i)17-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)21-s − i·23-s − i·27-s − 29-s − 31-s + (−0.866 − 0.5i)33-s + (−0.5 − 0.866i)39-s + (−0.5 + 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $-0.993 - 0.116i$
Analytic conductor: \(3.43654\)
Root analytic conductor: \(3.43654\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 740,\ (0:\ ),\ -0.993 - 0.116i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01151412322 + 0.1963851233i\)
\(L(\frac12)\) \(\approx\) \(0.01151412322 + 0.1963851233i\)
\(L(1)\) \(\approx\) \(0.8490367747 + 0.1535860354i\)
\(L(1)\) \(\approx\) \(0.8490367747 + 0.1535860354i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 - iT \)
29 \( 1 - T \)
31 \( 1 - T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 - iT \)
47 \( 1 - iT \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + iT \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 - iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.86452892316329532307821919669, −21.36145626386489319860627753913, −20.174214223231859823446306992416, −19.71129271489440812209919841470, −18.8495433583972090584376534766, −18.28014081327289386524652733618, −17.32212033547222267362903637667, −16.17264574220864544437975052671, −15.38604183811914267377364596520, −14.83099165879708378202710041929, −13.56895947341130844339894415325, −13.17670890002933113545265255384, −12.35723591461101785301639657242, −11.39487375123030814301171619322, −10.16769683651151574009566920590, −9.254558359469546483526590031071, −8.8119160189546950614529046467, −7.48414450243219645100095835478, −7.03985773837980137437590224203, −5.91022620340710836326111056601, −4.78847956900267905669971827331, −3.5616019698465884672063464112, −2.617525868602031164105940353668, −1.982397628204749922595954333993, −0.0719408017556741167665966440, 1.984768424479447829842371118539, 2.85208448350356401480029990290, 3.7817896092054458812051818133, 4.65594077074697915318063342813, 5.76323011051085963025925740215, 6.98699018435170392951186350149, 7.794495166807965459257762301497, 8.635402242842806334208065614140, 9.63057258130085999336899649159, 10.32579356476281678997245856162, 10.90076076627341590451659924315, 12.61871664406917246381534342499, 12.95611560719984461743558131428, 13.89905871417217446732356828144, 14.86638515607274888083313000411, 15.4148274335999798127592284454, 16.39645556716158710952318018132, 16.93929902121253629914953995872, 18.25214645237807438775295065351, 19.00294519319450975780427732024, 19.85166166472897682356502646411, 20.365358773470599746939957430119, 21.18307332298047997930754482975, 22.13777002223487393273180931710, 22.6727377300771958867739020538

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