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Properties

Label 1-3751-3751.1065-r1-0-0
Degree $1$
Conductor $3751$
Sign $0.799 + 0.600i$
Analytic cond. $403.100$
Root an. cond. $403.100$
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 + (−0.913 − 0.406i)3-s + (−0.959 + 0.281i)4-s + (−0.948 − 0.318i)5-s + (−0.272 + 0.962i)6-s + (0.0475 − 0.998i)7-s + (0.415 + 0.909i)8-s + (0.669 + 0.743i)9-s + (−0.179 + 0.983i)10-s + (0.991 + 0.132i)12-s + (−0.997 + 0.0760i)13-s + (−0.995 + 0.0950i)14-s + (0.736 + 0.676i)15-s + (0.841 − 0.540i)16-s + (−0.820 + 0.572i)17-s + (0.640 − 0.768i)18-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)2-s + (−0.913 − 0.406i)3-s + (−0.959 + 0.281i)4-s + (−0.948 − 0.318i)5-s + (−0.272 + 0.962i)6-s + (0.0475 − 0.998i)7-s + (0.415 + 0.909i)8-s + (0.669 + 0.743i)9-s + (−0.179 + 0.983i)10-s + (0.991 + 0.132i)12-s + (−0.997 + 0.0760i)13-s + (−0.995 + 0.0950i)14-s + (0.736 + 0.676i)15-s + (0.841 − 0.540i)16-s + (−0.820 + 0.572i)17-s + (0.640 − 0.768i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.799 + 0.600i$
Analytic conductor: \(403.100\)
Root analytic conductor: \(403.100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (1065, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (1:\ ),\ 0.799 + 0.600i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1919913510 - 0.06403123195i\)
\(L(\frac12)\) \(\approx\) \(-0.1919913510 - 0.06403123195i\)
\(L(1)\) \(\approx\) \(0.3217115548 - 0.3294948673i\)
\(L(1)\) \(\approx\) \(0.3217115548 - 0.3294948673i\)

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 \( 1 + (-0.913 - 0.406i)T \)
5 \( 1 + (-0.948 - 0.318i)T \)
7 \( 1 + (0.0475 - 0.998i)T \)
13 \( 1 + (-0.997 + 0.0760i)T \)
17 \( 1 + (-0.820 + 0.572i)T \)
19 \( 1 + (0.123 + 0.992i)T \)
23 \( 1 + (-0.774 - 0.633i)T \)
29 \( 1 + (-0.974 + 0.226i)T \)
37 \( 1 + (0.851 - 0.524i)T \)
41 \( 1 + (-0.0665 + 0.997i)T \)
43 \( 1 + (-0.00951 - 0.999i)T \)
47 \( 1 + (-0.466 - 0.884i)T \)
53 \( 1 + (-0.161 - 0.986i)T \)
59 \( 1 + (-0.0665 - 0.997i)T \)
61 \( 1 + (0.466 + 0.884i)T \)
67 \( 1 + (0.928 + 0.371i)T \)
71 \( 1 + (-0.290 + 0.956i)T \)
73 \( 1 + (-0.0475 - 0.998i)T \)
79 \( 1 + (0.217 - 0.976i)T \)
83 \( 1 + (-0.161 - 0.986i)T \)
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

−18.786901295734768896454847979523, −18.17519270616845399168846375347, −17.65261649671270847858204506613, −16.96487563512854605688287616764, −16.105984935402914473530726394393, −15.70975230441701531806234973660, −15.15053496207062949370113370734, −14.72340209560406381034276562867, −13.66835970232677160788991362454, −12.71915597931141471668271587749, −12.1622628837425843920034358470, −11.376485447170289032356209364092, −10.89687244153244864151483769347, −9.620592475761917588399311326005, −9.46898818571661158040349834525, −8.46224535269661917015634233170, −7.62361419854234885829329913217, −7.04368529813367240963255777966, −6.31900582699869570229600788244, −5.57690611659262661677624202131, −4.82183066350793729299755539652, −4.3896267918998092097485388862, −3.39688816297586772312823614323, −2.33808242564323253771003757457, −0.85864305267706534104964580385, 0.10306664185354099396666026597, 0.42856544442693435612602325632, 1.51056651712159246455914680057, 2.17171382092140309087114084935, 3.4995157660033389034532485900, 4.14714255502521289892642023079, 4.65970206989339638752908296227, 5.458021437404751561937964941781, 6.554981034567852382748358231088, 7.42785351383004568001657546022, 7.89238859206947733884006085074, 8.656696734456702417426324524402, 9.82250710491465054168326704284, 10.28549982047109025776285726688, 11.05427961413066932356835272182, 11.55609668241050332245604068804, 12.19654033149685437949912098755, 12.85617982949007788317000360310, 13.25924912233828399807283760113, 14.314242791651131924570875860690, 14.91656483760967409512897203405, 16.220947521508911970791724386452, 16.56826892077247413967838136989, 17.24814450937830323019717530819, 17.83844677911760095688671506269

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