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Properties

Label 2-3751-341.92-c0-0-4
Degree $2$
Conductor $3751$
Sign $0.970 - 0.242i$
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  + (−1.40 − 1.01i)2-s + (0.618 + 1.90i)4-s + (0.809 − 0.587i)5-s + (0.535 + 1.64i)7-s + (0.535 − 1.64i)8-s + (−0.809 − 0.587i)9-s − 1.73·10-s + (0.927 − 2.85i)14-s + (−0.809 + 0.587i)16-s + (0.535 + 1.64i)18-s + (−0.535 + 1.64i)19-s + (1.61 + 1.17i)20-s + (−2.80 + 2.03i)28-s + (0.809 + 0.587i)31-s + (1.40 + 1.01i)35-s + (0.618 − 1.90i)36-s + ⋯
L(s)  = 1  + (−1.40 − 1.01i)2-s + (0.618 + 1.90i)4-s + (0.809 − 0.587i)5-s + (0.535 + 1.64i)7-s + (0.535 − 1.64i)8-s + (−0.809 − 0.587i)9-s − 1.73·10-s + (0.927 − 2.85i)14-s + (−0.809 + 0.587i)16-s + (0.535 + 1.64i)18-s + (−0.535 + 1.64i)19-s + (1.61 + 1.17i)20-s + (−2.80 + 2.03i)28-s + (0.809 + 0.587i)31-s + (1.40 + 1.01i)35-s + (0.618 − 1.90i)36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.970 - 0.242i$
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: \((2,\ 3751,\ (\ :0),\ 0.970 - 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6345986640\)
\(L(\frac12)\) \(\approx\) \(0.6345986640\)
\(L(1)\) 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 \( 1 + (1.40 + 1.01i)T + (0.309 + 0.951i)T^{2} \)
3 \( 1 + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (-0.535 - 1.64i)T + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.535 - 1.64i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.535 + 1.64i)T + (-0.809 - 0.587i)T^{2} \)
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

−8.936820598108857011038521102311, −8.302588877921286383729202625129, −7.85894084442959952010750319803, −6.42418836423329217658961449576, −5.76444895518419845895684171301, −5.12785392892484194511481306787, −3.71408830843398440911656233467, −2.72280924965996263969254149150, −2.07569922146852978140872121071, −1.29529551624338625621673305677, 0.60784486415046477133004991480, 1.83401187027954727115620945304, 2.84622676423438866821377833543, 4.31502272736247905218838822685, 5.14503307762853046966170796817, 6.06710692494691351950645116476, 6.75136668097471402435084712749, 7.15749124876104216449666753112, 8.039034842801785065152782019965, 8.394626152332640779432923798448

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