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Properties

Label 2-1407-1407.1178-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.263 + 0.964i$
Analytic cond. $0.702184$
Root an. cond. $0.837964$
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  + (−0.327 + 0.945i)3-s + (−0.959 − 0.281i)4-s + (−0.995 + 0.0950i)7-s + (−0.786 − 0.618i)9-s + (0.580 − 0.814i)12-s + (−0.165 + 0.231i)13-s + (0.841 + 0.540i)16-s + (−0.279 − 1.94i)19-s + (0.235 − 0.971i)21-s + (0.580 − 0.814i)25-s + (0.841 − 0.540i)27-s + (0.981 + 0.189i)28-s + (0.698 − 1.53i)31-s + (0.580 + 0.814i)36-s − 1.99·37-s + ⋯
L(s)  = 1  + (−0.327 + 0.945i)3-s + (−0.959 − 0.281i)4-s + (−0.995 + 0.0950i)7-s + (−0.786 − 0.618i)9-s + (0.580 − 0.814i)12-s + (−0.165 + 0.231i)13-s + (0.841 + 0.540i)16-s + (−0.279 − 1.94i)19-s + (0.235 − 0.971i)21-s + (0.580 − 0.814i)25-s + (0.841 − 0.540i)27-s + (0.981 + 0.189i)28-s + (0.698 − 1.53i)31-s + (0.580 + 0.814i)36-s − 1.99·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.263 + 0.964i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.263 + 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3837730251\)
\(L(\frac12)\) \(\approx\) \(0.3837730251\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.327 - 0.945i)T \)
7 \( 1 + (0.995 - 0.0950i)T \)
67 \( 1 + (0.995 + 0.0950i)T \)
good2 \( 1 + (0.959 + 0.281i)T^{2} \)
5 \( 1 + (-0.580 + 0.814i)T^{2} \)
11 \( 1 + (-0.415 - 0.909i)T^{2} \)
13 \( 1 + (0.165 - 0.231i)T + (-0.327 - 0.945i)T^{2} \)
17 \( 1 + (0.888 + 0.458i)T^{2} \)
19 \( 1 + (0.279 + 1.94i)T + (-0.959 + 0.281i)T^{2} \)
23 \( 1 + (0.142 - 0.989i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2} \)
37 \( 1 + 1.99T + T^{2} \)
41 \( 1 + (0.888 + 0.458i)T^{2} \)
43 \( 1 + (0.797 - 0.234i)T + (0.841 - 0.540i)T^{2} \)
47 \( 1 + (0.786 + 0.618i)T^{2} \)
53 \( 1 + (0.888 - 0.458i)T^{2} \)
59 \( 1 + (-0.981 + 0.189i)T^{2} \)
61 \( 1 + (-1.65 + 1.06i)T + (0.415 - 0.909i)T^{2} \)
71 \( 1 + (-0.0475 + 0.998i)T^{2} \)
73 \( 1 + (0.0845 + 1.77i)T + (-0.995 + 0.0950i)T^{2} \)
79 \( 1 + (1.15 - 1.62i)T + (-0.327 - 0.945i)T^{2} \)
83 \( 1 + (0.995 + 0.0950i)T^{2} \)
89 \( 1 + (0.786 - 0.618i)T^{2} \)
97 \( 1 + (-0.327 - 0.566i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.613060208697036136726827556228, −8.982087706192314357607012838380, −8.385471947897249656896384057272, −6.90419497929353247505657822055, −6.20634240962714692982049780298, −5.21358592570826307744030786673, −4.57462821966293527359461672341, −3.71493581254899687449901780925, −2.69492885971146456884973667558, −0.35593572253296530114939293035, 1.40366092809216458476787613127, 3.00901398676428722671148130508, 3.78302619053693853500492874147, 5.10748165250362297987892966261, 5.77122003567802969675195210149, 6.75087935799171365502388783814, 7.43220676526913692092805978698, 8.430763397651976331510156257372, 8.840202113725965571419864867487, 10.11321407049322709204370757516

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