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Properties

Label 1-3751-3751.796-r1-0-0
Degree $1$
Conductor $3751$
Sign $0.507 - 0.861i$
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  + (0.941 − 0.336i)2-s + (−0.913 − 0.406i)3-s + (0.774 − 0.633i)4-s + (−0.999 + 0.0190i)5-s + (−0.997 − 0.0760i)6-s + (−0.217 + 0.976i)7-s + (0.516 − 0.856i)8-s + (0.669 + 0.743i)9-s + (−0.935 + 0.353i)10-s + (−0.964 + 0.263i)12-s + (0.888 − 0.458i)13-s + (0.123 + 0.992i)14-s + (0.921 + 0.389i)15-s + (0.198 − 0.980i)16-s + (−0.272 + 0.962i)17-s + (0.879 + 0.475i)18-s + ⋯
L(s)  = 1  + (0.941 − 0.336i)2-s + (−0.913 − 0.406i)3-s + (0.774 − 0.633i)4-s + (−0.999 + 0.0190i)5-s + (−0.997 − 0.0760i)6-s + (−0.217 + 0.976i)7-s + (0.516 − 0.856i)8-s + (0.669 + 0.743i)9-s + (−0.935 + 0.353i)10-s + (−0.964 + 0.263i)12-s + (0.888 − 0.458i)13-s + (0.123 + 0.992i)14-s + (0.921 + 0.389i)15-s + (0.198 − 0.980i)16-s + (−0.272 + 0.962i)17-s + (0.879 + 0.475i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.507 - 0.861i$
Analytic conductor: \(403.100\)
Root analytic conductor: \(403.100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (796, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (1:\ ),\ 0.507 - 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.745790165 - 0.9984946119i\)
\(L(\frac12)\) \(\approx\) \(1.745790165 - 0.9984946119i\)
\(L(1)\) \(\approx\) \(1.115035021 - 0.3136074514i\)
\(L(1)\) \(\approx\) \(1.115035021 - 0.3136074514i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.941 - 0.336i)T \)
3 \( 1 + (-0.913 - 0.406i)T \)
5 \( 1 + (-0.999 + 0.0190i)T \)
7 \( 1 + (-0.217 + 0.976i)T \)
13 \( 1 + (0.888 - 0.458i)T \)
17 \( 1 + (-0.272 + 0.962i)T \)
19 \( 1 + (-0.0665 - 0.997i)T \)
23 \( 1 + (-0.198 + 0.980i)T \)
29 \( 1 + (-0.696 + 0.717i)T \)
37 \( 1 + (-0.988 + 0.151i)T \)
41 \( 1 + (-0.432 - 0.901i)T \)
43 \( 1 + (0.999 + 0.0190i)T \)
47 \( 1 + (0.610 - 0.791i)T \)
53 \( 1 + (-0.953 + 0.299i)T \)
59 \( 1 + (-0.432 + 0.901i)T \)
61 \( 1 + (0.0285 - 0.999i)T \)
67 \( 1 + (0.723 - 0.690i)T \)
71 \( 1 + (0.999 + 0.0380i)T \)
73 \( 1 + (0.398 - 0.917i)T \)
79 \( 1 + (-0.483 + 0.875i)T \)
83 \( 1 + (-0.580 - 0.814i)T \)
89 \( 1 + (-0.696 - 0.717i)T \)
97 \( 1 + (0.0855 + 0.996i)T \)
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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

−18.55438205437230640005895658440, −17.62846409957122248954200444185, −16.75845975078487322322907246502, −16.44976914250542795736257461238, −15.837016923220659132541461770374, −15.3688752431693142137934394193, −14.34998417949101663539176528510, −13.90040766823385744101276092087, −12.84525983187109129329746290468, −12.45312460739514219874923205699, −11.54095878169413740509025268487, −11.181295978745919112382850862996, −10.56371758748577400165803882759, −9.658341626935884499257340971558, −8.53267669496491125573047369152, −7.73855163164278596888589534341, −7.011826385655340390752418922165, −6.502536664661827481432720417108, −5.72597185839029589728070622295, −4.79555757502606262913157921561, −4.14446539502371389431234697608, −3.83557872585392252995369783870, −2.94649078145592393737361303580, −1.53467847655136650743271473481, −0.52980533225292677282761487509, 0.42395383378799254682120963467, 1.43436989295278098141847752553, 2.22358383612082192463998846868, 3.313198179224063139741534166589, 3.8583649997080176589847811922, 4.84390843653097007726739106391, 5.45120202126903321024956280543, 6.09745042711811426719880841770, 6.80838401846664011847421049040, 7.50154440629565488536209383172, 8.40512226082531949813052249332, 9.27454525977894756681389231191, 10.473393652605517377850237911272, 10.98893491496511083399120180509, 11.464674362081859226765529758619, 12.246161426984931364338905920251, 12.62443786589362344691024745639, 13.25823525324218000795266481248, 14.056886396127513275918222790579, 15.24406374951671539951204708450, 15.49613612115839309612540824682, 15.93239567370259303463974999882, 16.86393471359971866373190385076, 17.69137946034524936000527659536, 18.60259233636887181526553390367

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