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Properties

Label 2-3751-341.193-c0-0-0
Degree $2$
Conductor $3751$
Sign $-0.0435 + 0.999i$
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  + (−1.47 − 0.658i)3-s + (0.309 − 0.951i)4-s + (0.895 + 0.994i)5-s + (1.08 + 1.20i)9-s + (−1.08 + 1.20i)12-s + (−0.669 − 2.05i)15-s + (−0.809 − 0.587i)16-s + (1.22 − 0.544i)20-s + (−1.47 − 1.07i)23-s + (−0.0826 + 0.786i)25-s + (−0.309 − 0.951i)27-s + (0.669 − 0.743i)31-s + (1.47 − 0.658i)36-s + (−0.913 − 0.406i)37-s + (−0.226 + 2.15i)45-s + ⋯
L(s)  = 1  + (−1.47 − 0.658i)3-s + (0.309 − 0.951i)4-s + (0.895 + 0.994i)5-s + (1.08 + 1.20i)9-s + (−1.08 + 1.20i)12-s + (−0.669 − 2.05i)15-s + (−0.809 − 0.587i)16-s + (1.22 − 0.544i)20-s + (−1.47 − 1.07i)23-s + (−0.0826 + 0.786i)25-s + (−0.309 − 0.951i)27-s + (0.669 − 0.743i)31-s + (1.47 − 0.658i)36-s + (−0.913 − 0.406i)37-s + (−0.226 + 2.15i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0435 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0435 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $-0.0435 + 0.999i$
Analytic conductor: \(1.87199\)
Root analytic conductor: \(1.36820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (3603, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3751,\ (\ :0),\ -0.0435 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8672241942\)
\(L(\frac12)\) \(\approx\) \(0.8672241942\)
\(L(1)\) 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 \( 1 + (-0.669 + 0.743i)T \)
good2 \( 1 + (-0.309 + 0.951i)T^{2} \)
3 \( 1 + (1.47 + 0.658i)T + (0.669 + 0.743i)T^{2} \)
5 \( 1 + (-0.895 - 0.994i)T + (-0.104 + 0.994i)T^{2} \)
7 \( 1 + (-0.913 + 0.406i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.669 - 0.743i)T^{2} \)
19 \( 1 + (-0.913 + 0.406i)T^{2} \)
23 \( 1 + (1.47 + 1.07i)T + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.913 + 0.406i)T + (0.669 + 0.743i)T^{2} \)
41 \( 1 + (0.104 - 0.994i)T^{2} \)
43 \( 1 + (0.104 + 0.994i)T^{2} \)
47 \( 1 + (-1.58 + 1.14i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.895 - 0.994i)T + (-0.104 + 0.994i)T^{2} \)
59 \( 1 + (-1.22 + 1.35i)T + (-0.104 - 0.994i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.978 + 1.69i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \)
73 \( 1 + (0.104 - 0.994i)T^{2} \)
79 \( 1 + (-0.913 + 0.406i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (1.47 + 1.07i)T + (0.309 + 0.951i)T^{2} \)
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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

−8.433594419173350579655174123770, −7.27310963700985137575509897517, −6.78643762895163938450061056324, −6.21131154928731964409303070326, −5.75417249335038709444750229445, −5.16936259219601432421937292896, −4.06800441548950109279999110325, −2.43750831478540714657671665761, −1.94714719000147212602564446520, −0.64775475318045356615430442453, 1.22076096369133740781749127975, 2.41112124903698540044550811918, 3.79405440599051485758178237433, 4.36103809470488948540680861125, 5.31126849215280340754808535849, 5.68780695775877818384188943353, 6.49144048200088343713829969774, 7.26604064960419718283114310174, 8.246722176620973905975570901137, 8.935142261828277687515519661111

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