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Properties

Label 2-3751-341.92-c0-0-14
Degree $2$
Conductor $3751$
Sign $0.394 + 0.918i$
Analytic cond. $1.87199$
Root an. cond. $1.36820$
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  + (0.809 + 0.587i)2-s + (0.809 − 0.587i)5-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + 10-s + (0.309 − 0.951i)14-s + (0.809 − 0.587i)16-s + (−0.309 − 0.951i)18-s + (−0.309 + 0.951i)19-s + (−0.809 − 0.587i)31-s + (−0.809 − 0.587i)35-s + (−0.809 + 0.587i)38-s + (−0.309 − 0.951i)40-s + (−0.309 + 0.951i)41-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.809 − 0.587i)5-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + 10-s + (0.309 − 0.951i)14-s + (0.809 − 0.587i)16-s + (−0.309 − 0.951i)18-s + (−0.309 + 0.951i)19-s + (−0.809 − 0.587i)31-s + (−0.809 − 0.587i)35-s + (−0.809 + 0.587i)38-s + (−0.309 − 0.951i)40-s + (−0.309 + 0.951i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.394 + 0.918i$
Analytic conductor: \(1.87199\)
Root analytic conductor: \(1.36820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (2138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3751,\ (\ :0),\ 0.394 + 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.878294047\)
\(L(\frac12)\) \(\approx\) \(1.878294047\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
3 \( 1 + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.523811297572916222778990718099, −7.65317006439107602398218689116, −6.77078528999218754723943059867, −6.23832280175916771384638210682, −5.54267293434978380573326131759, −5.01203950845598792244310935116, −3.92376382026510653570868208890, −3.49604723019356014815660060017, −1.98825885285986777616926286892, −0.794832609025574265063446231271, 1.97998059310487040622668083219, 2.58909926273536046115443110364, 3.07806847147647289718962838373, 4.18788539677489518681182815094, 5.17014741286682485294537916042, 5.62914948964720896353937230977, 6.34407447355327976325286589067, 7.27986102353502263297957841015, 8.232320479888020984348650656612, 8.865759035976756499585804414560

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