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Properties

Label 1-3751-3751.1047-r0-0-0
Degree $1$
Conductor $3751$
Sign $0.511 - 0.859i$
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 + (0.00951 + 0.999i)5-s + (0.999 + 0.0380i)6-s + (−0.0475 − 0.998i)7-s + (−0.415 + 0.909i)8-s + (−0.978 + 0.207i)9-s + (0.991 + 0.132i)10-s + (0.179 − 0.983i)12-s + (−0.851 + 0.524i)13-s + (−0.995 − 0.0950i)14-s + (−0.993 + 0.113i)15-s + (0.841 + 0.540i)16-s + (−0.290 + 0.956i)17-s + (0.0665 + 0.997i)18-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)2-s + (0.104 + 0.994i)3-s + (−0.959 − 0.281i)4-s + (0.00951 + 0.999i)5-s + (0.999 + 0.0380i)6-s + (−0.0475 − 0.998i)7-s + (−0.415 + 0.909i)8-s + (−0.978 + 0.207i)9-s + (0.991 + 0.132i)10-s + (0.179 − 0.983i)12-s + (−0.851 + 0.524i)13-s + (−0.995 − 0.0950i)14-s + (−0.993 + 0.113i)15-s + (0.841 + 0.540i)16-s + (−0.290 + 0.956i)17-s + (0.0665 + 0.997i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.511 - 0.859i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (1047, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ 0.511 - 0.859i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7947180633 - 0.4515762338i\)
\(L(\frac12)\) \(\approx\) \(0.7947180633 - 0.4515762338i\)
\(L(1)\) \(\approx\) \(0.8421770270 - 0.1059938038i\)
\(L(1)\) \(\approx\) \(0.8421770270 - 0.1059938038i\)

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 \( 1 + (0.00951 + 0.999i)T \)
7 \( 1 + (-0.0475 - 0.998i)T \)
13 \( 1 + (-0.851 + 0.524i)T \)
17 \( 1 + (-0.290 + 0.956i)T \)
19 \( 1 + (-0.483 - 0.875i)T \)
23 \( 1 + (0.254 - 0.967i)T \)
29 \( 1 + (0.0855 + 0.996i)T \)
37 \( 1 + (-0.997 + 0.0760i)T \)
41 \( 1 + (-0.640 - 0.768i)T \)
43 \( 1 + (-0.948 - 0.318i)T \)
47 \( 1 + (0.696 + 0.717i)T \)
53 \( 1 + (0.710 - 0.703i)T \)
59 \( 1 + (0.640 - 0.768i)T \)
61 \( 1 + (0.696 + 0.717i)T \)
67 \( 1 + (0.928 - 0.371i)T \)
71 \( 1 + (0.820 - 0.572i)T \)
73 \( 1 + (0.0475 - 0.998i)T \)
79 \( 1 + (-0.398 + 0.917i)T \)
83 \( 1 + (-0.710 + 0.703i)T \)
89 \( 1 + (-0.516 - 0.856i)T \)
97 \( 1 + (0.198 + 0.980i)T \)
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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

−18.56000995040271490106573886026, −17.98033473382014811945319183160, −17.17896602458573696967666685825, −16.91775952382774577073883575668, −15.88553667400211343121484752128, −15.36740687454787140517779454465, −14.65086980770604569439565459968, −13.8713359220351939064336927032, −13.19223584781881944415606713534, −12.72817011707014512789391843129, −11.9736791976717858413342648189, −11.65912812883166483577045360131, −9.9689006875961739467815387625, −9.416901991685316131799270422252, −8.54465815971383080251873810571, −8.27261935048522546730138077488, −7.43491965909261215797544407998, −6.779359017384404952804366864800, −5.78768298704236585318607257795, −5.44911231121141381693336841416, −4.75416970558414825743164574339, −3.64097781951990732224579131001, −2.66999726317379474367011197941, −1.78894316876882828637109387635, −0.68078891902516788010778506358, 0.35450622858311222324304442815, 1.88079967378865843749369941870, 2.54954844765630795277914299892, 3.41765229634510222508458899433, 3.92473512345674622788298156039, 4.64641762266273178906448008937, 5.312727684618154679309762366834, 6.49928568534312662467251296798, 7.07933416791360316856940555768, 8.27816014638185472625560193387, 8.87628124089558869173155681986, 9.80493476750774243401113208511, 10.29174690954199086495888000918, 10.79522246925986506088638091698, 11.2480402828063740920725290662, 12.14034798908660740276310396272, 13.00092100388359993301889957978, 13.859595086931489053742119673712, 14.31591372154290079439633931625, 14.89143523331757683375020840697, 15.51833959692414692833837394620, 16.668801279330599971139423147652, 17.21108290245865173851025450999, 17.73831817897853572551730576783, 18.745894573956606964255207258226

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