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Properties

Label 2-3751-341.48-c0-0-0
Degree $2$
Conductor $3751$
Sign $-0.153 + 0.988i$
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  + (−1.01 + 0.587i)3-s + (−0.309 + 0.951i)4-s + (0.204 + 1.94i)5-s + (0.190 − 0.330i)9-s + (−0.244 − 1.14i)12-s + (−1.35 − 1.86i)15-s + (−0.809 − 0.587i)16-s + (−1.91 − 0.406i)20-s + (−1.16 − 1.60i)23-s + (−2.76 + 0.587i)25-s − 0.726i·27-s + (0.104 − 0.994i)31-s + (0.255 + 0.283i)36-s + (1.28 + 1.15i)37-s + (0.682 + 0.303i)45-s + ⋯
L(s)  = 1  + (−1.01 + 0.587i)3-s + (−0.309 + 0.951i)4-s + (0.204 + 1.94i)5-s + (0.190 − 0.330i)9-s + (−0.244 − 1.14i)12-s + (−1.35 − 1.86i)15-s + (−0.809 − 0.587i)16-s + (−1.91 − 0.406i)20-s + (−1.16 − 1.60i)23-s + (−2.76 + 0.587i)25-s − 0.726i·27-s + (0.104 − 0.994i)31-s + (0.255 + 0.283i)36-s + (1.28 + 1.15i)37-s + (0.682 + 0.303i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $-0.153 + 0.988i$
Analytic conductor: \(1.87199\)
Root analytic conductor: \(1.36820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (1412, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3751,\ (\ :0),\ -0.153 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3340708751\)
\(L(\frac12)\) \(\approx\) \(0.3340708751\)
\(L(1)\) 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 \( 1 + (-0.104 + 0.994i)T \)
good2 \( 1 + (0.309 - 0.951i)T^{2} \)
3 \( 1 + (1.01 - 0.587i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.204 - 1.94i)T + (-0.978 + 0.207i)T^{2} \)
7 \( 1 + (0.913 - 0.406i)T^{2} \)
13 \( 1 + (0.104 - 0.994i)T^{2} \)
17 \( 1 + (-0.913 + 0.406i)T^{2} \)
19 \( 1 + (-0.978 - 0.207i)T^{2} \)
23 \( 1 + (1.16 + 1.60i)T + (-0.309 + 0.951i)T^{2} \)
29 \( 1 - T^{2} \)
37 \( 1 + (-1.28 - 1.15i)T + (0.104 + 0.994i)T^{2} \)
41 \( 1 + (0.669 - 0.743i)T^{2} \)
43 \( 1 + (-0.913 + 0.406i)T^{2} \)
47 \( 1 + 1.33T + T^{2} \)
53 \( 1 + (-0.360 - 0.207i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.190 + 0.0850i)T + (0.669 + 0.743i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.669 - 1.15i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.669 - 0.743i)T + (-0.104 + 0.994i)T^{2} \)
73 \( 1 + (0.104 - 0.994i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.913 - 0.406i)T^{2} \)
89 \( 1 + (1.01 - 1.40i)T + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.0646 - 0.198i)T + (-0.809 + 0.587i)T^{2} \)
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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

−9.530622721037232086261323146723, −8.177742670475762933969496695191, −7.84232901240709809788574562989, −6.73451678755493157027826778368, −6.42220096146364983698381272481, −5.62721977923825089921538223278, −4.49332944339107508970422463583, −3.96304352708402819088727903033, −2.92777193587235204113310431573, −2.36260719906156834805746713273, 0.22804205306541381597574952949, 1.29571262580610324270210666998, 1.81315738861376229961466730702, 3.77813444264960267919807742781, 4.75245299313495147796194014156, 5.19615043236545175112998356639, 5.87597330128368907701290735638, 6.26043682773419657445499100055, 7.42907680690174275174330062769, 8.229811056189319970679141704576

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