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Properties

Label 1-3751-3751.2557-r1-0-0
Degree $1$
Conductor $3751$
Sign $0.938 + 0.345i$
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  + (0.516 − 0.856i)2-s + (0.809 − 0.587i)3-s + (−0.466 − 0.884i)4-s + (−0.362 − 0.931i)5-s + (−0.0855 − 0.996i)6-s + (0.610 − 0.791i)7-s + (−0.998 − 0.0570i)8-s + (0.309 − 0.951i)9-s + (−0.985 − 0.170i)10-s + (−0.897 − 0.441i)12-s + (−0.696 + 0.717i)13-s + (−0.362 − 0.931i)14-s + (−0.841 − 0.540i)15-s + (−0.564 + 0.825i)16-s + (−0.415 − 0.909i)17-s + (−0.654 − 0.755i)18-s + ⋯
L(s)  = 1  + (0.516 − 0.856i)2-s + (0.809 − 0.587i)3-s + (−0.466 − 0.884i)4-s + (−0.362 − 0.931i)5-s + (−0.0855 − 0.996i)6-s + (0.610 − 0.791i)7-s + (−0.998 − 0.0570i)8-s + (0.309 − 0.951i)9-s + (−0.985 − 0.170i)10-s + (−0.897 − 0.441i)12-s + (−0.696 + 0.717i)13-s + (−0.362 − 0.931i)14-s + (−0.841 − 0.540i)15-s + (−0.564 + 0.825i)16-s + (−0.415 − 0.909i)17-s + (−0.654 − 0.755i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.938 + 0.345i$
Analytic conductor: \(403.100\)
Root analytic conductor: \(403.100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (2557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (1:\ ),\ 0.938 + 0.345i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.454127356 - 0.2590925797i\)
\(L(\frac12)\) \(\approx\) \(-1.454127356 - 0.2590925797i\)
\(L(1)\) \(\approx\) \(0.5771685713 - 1.279537366i\)
\(L(1)\) \(\approx\) \(0.5771685713 - 1.279537366i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.516 - 0.856i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.362 - 0.931i)T \)
7 \( 1 + (0.610 - 0.791i)T \)
13 \( 1 + (-0.696 + 0.717i)T \)
17 \( 1 + (-0.415 - 0.909i)T \)
19 \( 1 + (0.415 - 0.909i)T \)
23 \( 1 + (-0.941 + 0.336i)T \)
29 \( 1 + (0.870 - 0.491i)T \)
37 \( 1 + (0.985 + 0.170i)T \)
41 \( 1 + (-0.654 - 0.755i)T \)
43 \( 1 + (0.254 + 0.967i)T \)
47 \( 1 + (0.0855 - 0.996i)T \)
53 \( 1 + (0.0285 - 0.999i)T \)
59 \( 1 + (-0.654 + 0.755i)T \)
61 \( 1 + (-0.974 + 0.226i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (-0.941 + 0.336i)T \)
79 \( 1 + (-0.774 + 0.633i)T \)
83 \( 1 + (-0.610 + 0.791i)T \)
89 \( 1 + (-0.198 + 0.980i)T \)
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

−18.90414852024602005035207049438, −18.385369704476935871215044131969, −17.692754576206442636460556079828, −16.92607978961234138719930332556, −15.9127125091379281253380816136, −15.58057534515978816713952792404, −14.90742910026011298003528824634, −14.42879428482611486729942038465, −14.081253338714137569502875174350, −13.02117096560028334666892001756, −12.30107786662498657223398096675, −11.62986791805570465345959627559, −10.61596646446707437041520033862, −10.03812528495063793264950749611, −9.10086173485089616135214933487, −8.29872718502777210392479787692, −7.90047229322350279302503252757, −7.30271361190127032497112249288, −6.18850348464790740267093462688, −5.66805631979704994016611616032, −4.63334754798393182917583260706, −4.165162501085152317437377175562, −3.15800903800607908101986306513, −2.74985055252871861173613677527, −1.80085535449421421416681891394, 0.170858761026422969691698247309, 0.85035380653014861169228950192, 1.63028486406539791019834289974, 2.35572781528673864280281645152, 3.211330418338202776684373626654, 4.24302687668098423496798552878, 4.492006529661423066003005841536, 5.339238628636362707146059752162, 6.509120874544076127030632975115, 7.27994062317983542717465735835, 7.99720381696129543337752368185, 8.79093047266722151750493243325, 9.448094011918645113482619160610, 9.97766903250799294065375653806, 11.12448689374644583701270612726, 11.85976066992265546027246812001, 12.09430014333203224866367746919, 13.143123781832305660212854008192, 13.636461488982187822475602745488, 14.0004542001562981996045522931, 14.84654314286841385418338308578, 15.52865882938737905158769595349, 16.35483069212069521500613697789, 17.36129734483877761888496508965, 17.94135309419834005213011388176

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