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Properties

Label 2-3751-341.247-c0-0-5
Degree $2$
Conductor $3751$
Sign $-0.990 - 0.138i$
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  + (0.309 + 0.951i)2-s + (−0.309 + 0.951i)5-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s − 0.999·10-s + (−0.809 − 0.587i)14-s + (−0.309 + 0.951i)16-s + (−0.809 + 0.587i)18-s + (−0.809 − 0.587i)19-s + (0.309 + 0.951i)31-s + (−0.309 − 0.951i)35-s + (0.309 − 0.951i)38-s + (−0.809 + 0.587i)40-s + (−0.809 − 0.587i)41-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.309 + 0.951i)5-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s − 0.999·10-s + (−0.809 − 0.587i)14-s + (−0.309 + 0.951i)16-s + (−0.809 + 0.587i)18-s + (−0.809 − 0.587i)19-s + (0.309 + 0.951i)31-s + (−0.309 − 0.951i)35-s + (0.309 − 0.951i)38-s + (−0.809 + 0.587i)40-s + (−0.809 − 0.587i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $-0.990 - 0.138i$
Analytic conductor: \(1.87199\)
Root analytic conductor: \(1.36820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (3657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3751,\ (\ :0),\ -0.990 - 0.138i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.353963765\)
\(L(\frac12)\) \(\approx\) \(1.353963765\)
\(L(1)\) 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 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
3 \( 1 + (-0.309 - 0.951i)T^{2} \)
5 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
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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

−8.701058390920303968110987906175, −8.189980117819710988992516308851, −7.24504949962538116519099653036, −6.81128247432932497033493510738, −6.31861533934357259093068551499, −5.35529795677303989672817467700, −4.79997606311949822150705248377, −3.67624682483074220333461290045, −2.72216739588345125031384659056, −1.92599520115765515993983048090, 0.68169912607888718613339467906, 1.68208873407973455978056305945, 2.90241564963806221096291873120, 3.73678868391412645445383440287, 4.17754510240933359065461175748, 4.99406453692623404059795355226, 6.29562406628143974526387470672, 6.70989610856336594924782670186, 7.68326797484682113612249840457, 8.370189126122962266881292587798

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