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Properties

Label 1-3751-3751.1094-r0-0-0
Degree $1$
Conductor $3751$
Sign $-0.0376 - 0.999i$
Analytic cond. $17.4195$
Root an. cond. $17.4195$
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  + (−0.142 + 0.989i)2-s + (−0.104 + 0.994i)3-s + (−0.959 − 0.281i)4-s + (0.749 + 0.662i)5-s + (−0.969 − 0.244i)6-s + (−0.888 − 0.458i)7-s + (0.415 − 0.909i)8-s + (−0.978 − 0.207i)9-s + (−0.761 + 0.647i)10-s + (0.380 − 0.924i)12-s + (−0.432 − 0.901i)13-s + (0.580 − 0.814i)14-s + (−0.736 + 0.676i)15-s + (0.841 + 0.540i)16-s + (−0.905 + 0.424i)17-s + (0.345 − 0.938i)18-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)2-s + (−0.104 + 0.994i)3-s + (−0.959 − 0.281i)4-s + (0.749 + 0.662i)5-s + (−0.969 − 0.244i)6-s + (−0.888 − 0.458i)7-s + (0.415 − 0.909i)8-s + (−0.978 − 0.207i)9-s + (−0.761 + 0.647i)10-s + (0.380 − 0.924i)12-s + (−0.432 − 0.901i)13-s + (0.580 − 0.814i)14-s + (−0.736 + 0.676i)15-s + (0.841 + 0.540i)16-s + (−0.905 + 0.424i)17-s + (0.345 − 0.938i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0376 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0376 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $-0.0376 - 0.999i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (1094, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ -0.0376 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3199807429 + 0.3322788836i\)
\(L(\frac12)\) \(\approx\) \(-0.3199807429 + 0.3322788836i\)
\(L(1)\) \(\approx\) \(0.4836748191 + 0.5732835424i\)
\(L(1)\) \(\approx\) \(0.4836748191 + 0.5732835424i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.142 + 0.989i)T \)
3 \( 1 + (-0.104 + 0.994i)T \)
5 \( 1 + (0.749 + 0.662i)T \)
7 \( 1 + (-0.888 - 0.458i)T \)
13 \( 1 + (-0.432 - 0.901i)T \)
17 \( 1 + (-0.905 + 0.424i)T \)
19 \( 1 + (0.797 + 0.603i)T \)
23 \( 1 + (0.774 - 0.633i)T \)
29 \( 1 + (0.974 + 0.226i)T \)
37 \( 1 + (0.879 - 0.475i)T \)
41 \( 1 + (-0.830 + 0.556i)T \)
43 \( 1 + (0.861 + 0.508i)T \)
47 \( 1 + (-0.466 + 0.884i)T \)
53 \( 1 + (-0.935 + 0.353i)T \)
59 \( 1 + (-0.830 - 0.556i)T \)
61 \( 1 + (-0.466 + 0.884i)T \)
67 \( 1 + (-0.786 - 0.618i)T \)
71 \( 1 + (-0.683 + 0.730i)T \)
73 \( 1 + (-0.888 + 0.458i)T \)
79 \( 1 + (0.953 + 0.299i)T \)
83 \( 1 + (-0.935 + 0.353i)T \)
89 \( 1 + (-0.921 - 0.389i)T \)
97 \( 1 + (-0.870 + 0.491i)T \)
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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

−18.182642548423770744757108213796, −17.6206866324650242874834103912, −16.98935167793750482703887322317, −16.36374676558254653739607692877, −15.37081765309606105328344015703, −14.17790362102056356398847877184, −13.644863090543984763524255587377, −13.21214613288947537993748024212, −12.598712202837168441020037268498, −11.86099187460629458844792019828, −11.51221297698327878851386264277, −10.442130704490134891070907879492, −9.570447901320317503938145091195, −9.084271369825739676476672151902, −8.64811226474054266254475422918, −7.53491879879029126133064785279, −6.74754617945004957509655093983, −6.00485358101533610333188900176, −5.12288252018268362115017997854, −4.56567764328065567580359881603, −3.22968210911521332335382736237, −2.6032481516884866791744332161, −1.94388580642277164947701987265, −1.13885824952804313179360912284, −0.15999317155386269372778849725, 1.148579259409083242964974320734, 2.80570105834619822675118490760, 3.21121388003145200708856188811, 4.27725470668952174728759951190, 4.895939441537727033161455540080, 5.87947306236461067320561844430, 6.19262330618204358231211345456, 7.01289721290592875807102136804, 7.81181687800359802021871331879, 8.75010086548413866390279554740, 9.46536341615285939110865596309, 9.96581430517895879918828956142, 10.49787844835878136686278068511, 11.12243160765282920060884336386, 12.52264623198810815154045279484, 13.09348965948016178840111881364, 13.951742471457707144772659531408, 14.41408821605034559452158334402, 15.16959372091053461376095976951, 15.65332888334998053067313448971, 16.43515811184777132093898577872, 16.91992264476802911421365092127, 17.663196105258301945620169529581, 18.05719322653912572905238838080, 19.05113906013757168172353846829

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