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Properties

Label 1-3751-3751.174-r0-0-0
Degree $1$
Conductor $3751$
Sign $0.938 + 0.344i$
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  + (0.516 + 0.856i)2-s + (0.913 − 0.406i)3-s + (−0.466 + 0.884i)4-s + (0.988 − 0.151i)5-s + (0.820 + 0.572i)6-s + (−0.991 + 0.132i)7-s + (−0.998 + 0.0570i)8-s + (0.669 − 0.743i)9-s + (0.640 + 0.768i)10-s + (−0.0665 + 0.997i)12-s + (0.272 − 0.962i)13-s + (−0.625 − 0.780i)14-s + (0.841 − 0.540i)15-s + (−0.564 − 0.825i)16-s + (0.580 + 0.814i)17-s + (0.981 + 0.189i)18-s + ⋯
L(s)  = 1  + (0.516 + 0.856i)2-s + (0.913 − 0.406i)3-s + (−0.466 + 0.884i)4-s + (0.988 − 0.151i)5-s + (0.820 + 0.572i)6-s + (−0.991 + 0.132i)7-s + (−0.998 + 0.0570i)8-s + (0.669 − 0.743i)9-s + (0.640 + 0.768i)10-s + (−0.0665 + 0.997i)12-s + (0.272 − 0.962i)13-s + (−0.625 − 0.780i)14-s + (0.841 − 0.540i)15-s + (−0.564 − 0.825i)16-s + (0.580 + 0.814i)17-s + (0.981 + 0.189i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.938 + 0.344i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ 0.938 + 0.344i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.313071036 + 0.5889393524i\)
\(L(\frac12)\) \(\approx\) \(3.313071036 + 0.5889393524i\)
\(L(1)\) \(\approx\) \(1.845494374 + 0.4937365736i\)
\(L(1)\) \(\approx\) \(1.845494374 + 0.4937365736i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.516 + 0.856i)T \)
3 \( 1 + (0.913 - 0.406i)T \)
5 \( 1 + (0.988 - 0.151i)T \)
7 \( 1 + (-0.991 + 0.132i)T \)
13 \( 1 + (0.272 - 0.962i)T \)
17 \( 1 + (0.580 + 0.814i)T \)
19 \( 1 + (-0.995 - 0.0950i)T \)
23 \( 1 + (0.941 + 0.336i)T \)
29 \( 1 + (-0.870 - 0.491i)T \)
37 \( 1 + (0.345 - 0.938i)T \)
41 \( 1 + (0.981 + 0.189i)T \)
43 \( 1 + (-0.710 - 0.703i)T \)
47 \( 1 + (0.0855 + 0.996i)T \)
53 \( 1 + (-0.851 + 0.524i)T \)
59 \( 1 + (0.981 - 0.189i)T \)
61 \( 1 + (0.974 + 0.226i)T \)
67 \( 1 + (0.981 + 0.189i)T \)
71 \( 1 + (0.580 - 0.814i)T \)
73 \( 1 + (-0.179 - 0.983i)T \)
79 \( 1 + (-0.935 + 0.353i)T \)
83 \( 1 + (0.380 - 0.924i)T \)
89 \( 1 + (0.198 + 0.980i)T \)
97 \( 1 + (0.841 + 0.540i)T \)
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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

−18.84308348450347959911052746615, −18.33508630425970245434006119441, −17.105667922786850418212312645490, −16.47637455921278507964379582163, −15.72212457742924525590272435549, −14.73439003846791261140548866593, −14.413365635634516391049054957129, −13.7090940288788342232162172143, −12.95173702706058465699553468185, −12.86677426937427387334390558094, −11.569366945141134787993565570916, −10.83987585425138007117783443117, −10.07694802886896451554086610950, −9.631916310590707203845120527403, −9.11273303723762124095149154914, −8.43361202587611437924446668766, −6.99320651646310709915720937901, −6.557727007966367954203583065449, −5.55547411953938456574711646439, −4.827895515988510651441576681701, −3.97584369499850412502882991480, −3.28666556733583439491462763542, −2.60404112588278986966481052680, −1.99054611990917315463335481658, −1.0478850796883471060218267095, 0.75970993171299251075676344254, 2.02474063667856420663460142697, 2.8045782501515710792313795448, 3.46055640089629149086143584481, 4.18795172994395349913154508202, 5.361823505495439491244137897006, 5.98884682385296648322290676643, 6.505509933436916868001490740456, 7.31408945488408466419166563719, 8.06203122033223340371505888135, 8.785277512273861911714978722911, 9.3620468352022614446409177667, 9.984444854337837060748790044814, 10.93875817905592134253026759397, 12.38638549269055879033076420769, 12.80822370909139407277820091392, 13.14574302922654564869596518589, 13.76969903533866144096350943245, 14.682318409639428090176075181178, 14.99378145631760936745021489551, 15.79670203476351686094621200491, 16.52697009106777836010344286830, 17.35149289687302769906766633122, 17.71046049417988657350295229257, 18.75640001506903108324120587225

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