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Properties

Label 1-3751-3751.2546-r0-0-0
Degree $1$
Conductor $3751$
Sign $0.177 + 0.984i$
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  + (0.897 + 0.441i)2-s + (0.309 − 0.951i)3-s + (0.610 + 0.791i)4-s + (0.198 − 0.980i)5-s + (0.696 − 0.717i)6-s + (−0.998 − 0.0570i)7-s + (0.198 + 0.980i)8-s + (−0.809 − 0.587i)9-s + (0.610 − 0.791i)10-s + (0.941 − 0.336i)12-s + (−0.959 − 0.281i)13-s + (−0.870 − 0.491i)14-s + (−0.870 − 0.491i)15-s + (−0.254 + 0.967i)16-s + (0.516 + 0.856i)17-s + (−0.466 − 0.884i)18-s + ⋯
L(s)  = 1  + (0.897 + 0.441i)2-s + (0.309 − 0.951i)3-s + (0.610 + 0.791i)4-s + (0.198 − 0.980i)5-s + (0.696 − 0.717i)6-s + (−0.998 − 0.0570i)7-s + (0.198 + 0.980i)8-s + (−0.809 − 0.587i)9-s + (0.610 − 0.791i)10-s + (0.941 − 0.336i)12-s + (−0.959 − 0.281i)13-s + (−0.870 − 0.491i)14-s + (−0.870 − 0.491i)15-s + (−0.254 + 0.967i)16-s + (0.516 + 0.856i)17-s + (−0.466 − 0.884i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.177 + 0.984i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (2546, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ 0.177 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.348652745 + 1.127746018i\)
\(L(\frac12)\) \(\approx\) \(1.348652745 + 1.127746018i\)
\(L(1)\) \(\approx\) \(1.507435746 + 0.01990084655i\)
\(L(1)\) \(\approx\) \(1.507435746 + 0.01990084655i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.897 + 0.441i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.198 - 0.980i)T \)
7 \( 1 + (-0.998 - 0.0570i)T \)
13 \( 1 + (-0.959 - 0.281i)T \)
17 \( 1 + (0.516 + 0.856i)T \)
19 \( 1 + (0.0855 + 0.996i)T \)
23 \( 1 + (-0.254 + 0.967i)T \)
29 \( 1 + (0.516 - 0.856i)T \)
37 \( 1 + (-0.0285 + 0.999i)T \)
41 \( 1 + (0.696 - 0.717i)T \)
43 \( 1 + (0.198 + 0.980i)T \)
47 \( 1 + (-0.985 - 0.170i)T \)
53 \( 1 + (-0.998 - 0.0570i)T \)
59 \( 1 + (0.696 + 0.717i)T \)
61 \( 1 + (-0.466 + 0.884i)T \)
67 \( 1 + (-0.142 + 0.989i)T \)
71 \( 1 + (-0.921 + 0.389i)T \)
73 \( 1 + (-0.362 - 0.931i)T \)
79 \( 1 + (0.198 - 0.980i)T \)
83 \( 1 + (0.841 - 0.540i)T \)
89 \( 1 + (0.516 + 0.856i)T \)
97 \( 1 + (0.993 + 0.113i)T \)
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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

−18.748739266916696856397258585448, −17.8678757372881085519429145609, −16.80782425614066998921401079586, −16.10199848393329206999466381681, −15.65735335663983066999882334899, −14.86893443611822888620319112928, −14.27382588567386477924403039883, −13.94539356873204928878083824792, −12.988826755359746231132002726007, −12.26689820449176729043994902858, −11.45403676162345146011989106892, −10.80190859869966038596892895590, −10.18320661885505543235641300735, −9.59539471214379350765273878763, −9.12263265295745867899039330144, −7.699466988968156904849947567662, −6.89603540083541084364286855469, −6.35845343023958883940370220751, −5.414735092340805194636457277760, −4.78838381581414463000999133474, −3.96437240076217478106772502715, −3.07743738135493797516891702405, −2.814993236168107137721263498292, −2.06848604947031907332250800377, −0.32119458119309165395208675498, 1.16694113342049196863078155565, 2.00777215614612317873282022193, 2.89428853892839198917957347142, 3.59768056832627354292291197459, 4.42766080404808656433646841230, 5.4704979262615113071806632177, 5.96740983840525040229117302249, 6.54556637241898726261674234741, 7.64080999918696470959834580163, 7.862108589096078080408545905254, 8.75900323572624037748826519430, 9.613146158668860618967590692087, 10.33970988340548500779943316281, 11.83211133454819827463305101061, 12.02397053268443292564246187473, 12.78329496929720853593362975300, 13.17837935632658515918978366755, 13.770206085018767730913067171746, 14.58567230019764581918419239202, 15.167614544227573626471265638793, 16.1525275302311875041170396430, 16.56302411557049841221096911374, 17.43846633239485945214907334982, 17.66844290856055068382985987725, 19.14385251494217636486854905803

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