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Properties

Label 1-3751-3751.746-r0-0-0
Degree $1$
Conductor $3751$
Sign $0.998 - 0.0615i$
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  + (0.774 − 0.633i)2-s + (−0.809 − 0.587i)3-s + (0.198 − 0.980i)4-s + (0.696 + 0.717i)5-s + (−0.998 + 0.0570i)6-s + (0.0855 + 0.996i)7-s + (−0.466 − 0.884i)8-s + (0.309 + 0.951i)9-s + (0.993 + 0.113i)10-s + (−0.736 + 0.676i)12-s + (−0.870 − 0.491i)13-s + (0.696 + 0.717i)14-s + (−0.142 − 0.989i)15-s + (−0.921 − 0.389i)16-s + (−0.959 + 0.281i)17-s + (0.841 + 0.540i)18-s + ⋯
L(s)  = 1  + (0.774 − 0.633i)2-s + (−0.809 − 0.587i)3-s + (0.198 − 0.980i)4-s + (0.696 + 0.717i)5-s + (−0.998 + 0.0570i)6-s + (0.0855 + 0.996i)7-s + (−0.466 − 0.884i)8-s + (0.309 + 0.951i)9-s + (0.993 + 0.113i)10-s + (−0.736 + 0.676i)12-s + (−0.870 − 0.491i)13-s + (0.696 + 0.717i)14-s + (−0.142 − 0.989i)15-s + (−0.921 − 0.389i)16-s + (−0.959 + 0.281i)17-s + (0.841 + 0.540i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.998 - 0.0615i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (746, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ 0.998 - 0.0615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.796229232 - 0.05529940062i\)
\(L(\frac12)\) \(\approx\) \(1.796229232 - 0.05529940062i\)
\(L(1)\) \(\approx\) \(1.204856963 - 0.3854134222i\)
\(L(1)\) \(\approx\) \(1.204856963 - 0.3854134222i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.774 - 0.633i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (0.696 + 0.717i)T \)
7 \( 1 + (0.0855 + 0.996i)T \)
13 \( 1 + (-0.870 - 0.491i)T \)
17 \( 1 + (-0.959 + 0.281i)T \)
19 \( 1 + (-0.959 - 0.281i)T \)
23 \( 1 + (0.974 - 0.226i)T \)
29 \( 1 + (0.941 - 0.336i)T \)
37 \( 1 + (0.993 + 0.113i)T \)
41 \( 1 + (0.841 + 0.540i)T \)
43 \( 1 + (-0.985 + 0.170i)T \)
47 \( 1 + (-0.998 - 0.0570i)T \)
53 \( 1 + (0.516 - 0.856i)T \)
59 \( 1 + (0.841 - 0.540i)T \)
61 \( 1 + (-0.362 + 0.931i)T \)
67 \( 1 + (0.841 + 0.540i)T \)
71 \( 1 + (-0.959 - 0.281i)T \)
73 \( 1 + (0.974 - 0.226i)T \)
79 \( 1 + (0.897 - 0.441i)T \)
83 \( 1 + (0.0855 + 0.996i)T \)
89 \( 1 + (0.610 - 0.791i)T \)
97 \( 1 + (-0.142 + 0.989i)T \)
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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

−18.13593335180908615395940436507, −17.57075640896115171126671183291, −16.976972507181847748175192936052, −16.69059103398537203839469183393, −16.03268106802226027292702439088, −15.20828277617494385298895867200, −14.52306960059497970741941060599, −13.85136609339068340303594548718, −13.05922177752171865919233968942, −12.628442277006943829923281538103, −11.795608082396560790705846159859, −11.069486551772019291938337115794, −10.37148450667286340060678281291, −9.50279175980287984589228365394, −8.88856843655849177134165184715, −7.9625272252036275765320693642, −6.86329742881752020118544232484, −6.62968251264620424800894001942, −5.70468949581384430615976252247, −4.86074614293851039592772710049, −4.570367553867221305595823966169, −3.911792731718334988748375967717, −2.775883500717235350464169825234, −1.73971168412439957239868856750, −0.49007951780381163309767279098, 0.90603910153943479707894316580, 2.15609752688998694773713344819, 2.29721668405641256587094805053, 3.14670944837324442790278545727, 4.53074462030118736957836834690, 5.04052878841887345881321148708, 5.77111027748836043776370974296, 6.5327799820211324744826244593, 6.74054736231748142733761413991, 7.96938736193099692224312586884, 8.99227321274354316339212492852, 9.87020180003473911032657619119, 10.4646509219240137098815260745, 11.19938991002457326262210536480, 11.62120059269299093462472204605, 12.511728064538599965416431334012, 13.043795056420213893556317274088, 13.44540308101209908066958714417, 14.64007999466692111456230734678, 14.87118027095305180109663973787, 15.6491080726809153478364648224, 16.62110656277708710154399044100, 17.572149576683788158215463657571, 17.94683650739126607013979461654, 18.611533958496458931099917259755

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