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Properties

Label 1-3751-3751.1106-r0-0-0
Degree $1$
Conductor $3751$
Sign $0.215 - 0.976i$
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  + (−0.610 − 0.791i)2-s + (0.5 + 0.866i)3-s + (−0.254 + 0.967i)4-s + (−0.290 + 0.956i)5-s + (0.380 − 0.924i)6-s + (0.398 + 0.917i)7-s + (0.921 − 0.389i)8-s + (−0.5 + 0.866i)9-s + (0.935 − 0.353i)10-s + (−0.964 + 0.263i)12-s + (−0.710 − 0.703i)13-s + (0.483 − 0.875i)14-s + (−0.974 + 0.226i)15-s + (−0.870 − 0.491i)16-s + (−0.830 − 0.556i)17-s + (0.991 − 0.132i)18-s + ⋯
L(s)  = 1  + (−0.610 − 0.791i)2-s + (0.5 + 0.866i)3-s + (−0.254 + 0.967i)4-s + (−0.290 + 0.956i)5-s + (0.380 − 0.924i)6-s + (0.398 + 0.917i)7-s + (0.921 − 0.389i)8-s + (−0.5 + 0.866i)9-s + (0.935 − 0.353i)10-s + (−0.964 + 0.263i)12-s + (−0.710 − 0.703i)13-s + (0.483 − 0.875i)14-s + (−0.974 + 0.226i)15-s + (−0.870 − 0.491i)16-s + (−0.830 − 0.556i)17-s + (0.991 − 0.132i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.215 - 0.976i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (1106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ 0.215 - 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3131844695 - 0.2515112650i\)
\(L(\frac12)\) \(\approx\) \(0.3131844695 - 0.2515112650i\)
\(L(1)\) \(\approx\) \(0.7031697298 + 0.1292448824i\)
\(L(1)\) \(\approx\) \(0.7031697298 + 0.1292448824i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.610 - 0.791i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.290 + 0.956i)T \)
7 \( 1 + (0.398 + 0.917i)T \)
13 \( 1 + (-0.710 - 0.703i)T \)
17 \( 1 + (-0.830 - 0.556i)T \)
19 \( 1 + (0.532 - 0.846i)T \)
23 \( 1 + (-0.198 + 0.980i)T \)
29 \( 1 + (-0.142 + 0.989i)T \)
37 \( 1 + (0.710 - 0.703i)T \)
41 \( 1 + (0.179 - 0.983i)T \)
43 \( 1 + (-0.999 - 0.0190i)T \)
47 \( 1 + (-0.959 + 0.281i)T \)
53 \( 1 + (-0.580 - 0.814i)T \)
59 \( 1 + (-0.179 - 0.983i)T \)
61 \( 1 + (0.941 + 0.336i)T \)
67 \( 1 + (0.723 - 0.690i)T \)
71 \( 1 + (0.345 - 0.938i)T \)
73 \( 1 + (-0.217 - 0.976i)T \)
79 \( 1 + (0.981 + 0.189i)T \)
83 \( 1 + (0.953 - 0.299i)T \)
89 \( 1 + (-0.696 - 0.717i)T \)
97 \( 1 + (-0.921 + 0.389i)T \)
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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

−18.7377231737065255026786297874, −17.98728991516561204292676614609, −17.31097243620017934660996804865, −16.78099177002983417838915838419, −16.30838897285525691177291851854, −15.25910230810288818703420570344, −14.66169908195166876655038594800, −14.028585815460548721610626887361, −13.3718421093916624271698672301, −12.79442112947835997382093843114, −11.80110584337883581714446370407, −11.25198030060885324075651580863, −10.0554452915547030004218641638, −9.56545444671448315712084200595, −8.679708617724581259636365861206, −8.05913789496486314874339752452, −7.76768633777057110254979145627, −6.80452027557491308163592513240, −6.35063994946643542784320788778, −5.29388282762919201184242925042, −4.44843146988124266974261430207, −3.91723607163177662234487042823, −2.4001017775136850155266918919, −1.55932769998355197786308548588, −0.966225727031795365340679752751, 0.14509125664142076840474118904, 1.89283192490908900734261904179, 2.47938640864132151808939704776, 3.13253391645295296001989011904, 3.68690528481520404871150008661, 4.82822270032879269629632657109, 5.25595123968535157387027071610, 6.617828548097091396098609908337, 7.560004257272517868709588288528, 7.992586329619223534804810706839, 8.927320701215299473990558475498, 9.433330808478027588683972683, 10.044889042112088802276323803695, 10.946886028457805949744854668826, 11.26459472237375078947269593442, 11.93013235683479761216968923736, 12.87541739988159015706005247987, 13.69731859042816415308229896807, 14.43357525775879411352340313903, 15.13959758698547437616614277502, 15.70510189525247214647617476622, 16.26367295772308357956461631953, 17.44599501423288783928065455927, 17.865720173130792862797471468940, 18.4549060410773720537764703977

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