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Properties

Label 1-3751-3751.784-r0-0-0
Degree $1$
Conductor $3751$
Sign $-0.920 - 0.390i$
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  + (0.0855 − 0.996i)2-s + (−0.978 + 0.207i)3-s + (−0.985 − 0.170i)4-s + (−0.935 + 0.353i)5-s + (0.123 + 0.992i)6-s + (0.997 + 0.0760i)7-s + (−0.254 + 0.967i)8-s + (0.913 − 0.406i)9-s + (0.272 + 0.962i)10-s + (0.999 − 0.0380i)12-s + (−0.0665 + 0.997i)13-s + (0.161 − 0.986i)14-s + (0.841 − 0.540i)15-s + (0.941 + 0.336i)16-s + (−0.995 + 0.0950i)17-s + (−0.327 − 0.945i)18-s + ⋯
L(s)  = 1  + (0.0855 − 0.996i)2-s + (−0.978 + 0.207i)3-s + (−0.985 − 0.170i)4-s + (−0.935 + 0.353i)5-s + (0.123 + 0.992i)6-s + (0.997 + 0.0760i)7-s + (−0.254 + 0.967i)8-s + (0.913 − 0.406i)9-s + (0.272 + 0.962i)10-s + (0.999 − 0.0380i)12-s + (−0.0665 + 0.997i)13-s + (0.161 − 0.986i)14-s + (0.841 − 0.540i)15-s + (0.941 + 0.336i)16-s + (−0.995 + 0.0950i)17-s + (−0.327 − 0.945i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $-0.920 - 0.390i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (784, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ -0.920 - 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05464311102 - 0.2688176259i\)
\(L(\frac12)\) \(\approx\) \(0.05464311102 - 0.2688176259i\)
\(L(1)\) \(\approx\) \(0.5606435503 - 0.1773123298i\)
\(L(1)\) \(\approx\) \(0.5606435503 - 0.1773123298i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.0855 - 0.996i)T \)
3 \( 1 + (-0.978 + 0.207i)T \)
5 \( 1 + (-0.935 + 0.353i)T \)
7 \( 1 + (0.997 + 0.0760i)T \)
13 \( 1 + (-0.0665 + 0.997i)T \)
17 \( 1 + (-0.995 + 0.0950i)T \)
19 \( 1 + (0.580 - 0.814i)T \)
23 \( 1 + (-0.0285 + 0.999i)T \)
29 \( 1 + (-0.736 + 0.676i)T \)
37 \( 1 + (-0.969 - 0.244i)T \)
41 \( 1 + (-0.327 - 0.945i)T \)
43 \( 1 + (-0.625 + 0.780i)T \)
47 \( 1 + (-0.921 + 0.389i)T \)
53 \( 1 + (-0.991 + 0.132i)T \)
59 \( 1 + (-0.327 + 0.945i)T \)
61 \( 1 + (0.516 - 0.856i)T \)
67 \( 1 + (-0.327 - 0.945i)T \)
71 \( 1 + (-0.995 - 0.0950i)T \)
73 \( 1 + (-0.851 - 0.524i)T \)
79 \( 1 + (0.449 + 0.893i)T \)
83 \( 1 + (-0.432 - 0.901i)T \)
89 \( 1 + (0.993 + 0.113i)T \)
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

−18.568600670495022411899273478792, −18.07901841293644774753372716763, −17.42144933838494759074118421768, −16.858701817416031979300289896754, −16.19492788458919263116097886596, −15.56233641541954200606192099315, −15.02737422663412913351945212907, −14.29700376891880830883248012827, −13.2599474858435468582476345198, −12.84316429569872421312437805982, −11.94841436002885651081973345567, −11.51624275220295394281571888172, −10.593937322034789637323462356998, −9.9465473386772115597323818641, −8.73938437788626487765871466472, −8.18990788028808344510009282248, −7.59632857912256599722041816089, −6.98678894990986432339137471405, −6.11767055992027419535967874298, −5.321227057226418257722625708353, −4.794210631913688012919474801378, −4.22159800191452634832029152515, −3.31580343694818136848492274768, −1.760561895207589415749355837465, −0.76895311657308942357594822448, 0.12863246554733121042292697606, 1.378768515781092703695542820752, 1.98873327950039407915777349985, 3.24532225076887590995084847305, 3.92561414850350923385863245752, 4.78988073327629674858564461084, 4.95531262079729601524403199218, 6.11730668793337534956024345575, 7.079134574349734783242989684906, 7.6820514564785293181001964052, 8.7901282792144223390050656501, 9.25728088765109918619361973233, 10.29038208362752015567084608775, 11.01559214266587575337348702068, 11.394952902765898133370280276067, 11.75563266505721867093090267117, 12.49648579656887228489446027015, 13.36134146124077141825467800196, 14.12259126360483513607020434898, 14.9152135124039640863281484227, 15.52472966559877403309946925605, 16.27922259856885673199074683571, 17.23772303921205982672075898330, 17.74808462249858059759690902398, 18.31713328639204751533751205820

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