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Properties

Label 2-1407-1407.677-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.292 + 0.956i$
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  + (−0.235 − 0.971i)3-s + (0.995 + 0.0950i)4-s + (0.841 − 0.540i)7-s + (−0.888 + 0.458i)9-s + (−0.142 − 0.989i)12-s + (−1.56 − 0.625i)13-s + (0.981 + 0.189i)16-s + (0.890 − 1.72i)19-s + (−0.723 − 0.690i)21-s + (−0.786 + 0.618i)25-s + (0.654 + 0.755i)27-s + (0.888 − 0.458i)28-s + (1.02 + 0.809i)31-s + (−0.928 + 0.371i)36-s + (0.786 + 1.36i)37-s + ⋯
L(s)  = 1  + (−0.235 − 0.971i)3-s + (0.995 + 0.0950i)4-s + (0.841 − 0.540i)7-s + (−0.888 + 0.458i)9-s + (−0.142 − 0.989i)12-s + (−1.56 − 0.625i)13-s + (0.981 + 0.189i)16-s + (0.890 − 1.72i)19-s + (−0.723 − 0.690i)21-s + (−0.786 + 0.618i)25-s + (0.654 + 0.755i)27-s + (0.888 − 0.458i)28-s + (1.02 + 0.809i)31-s + (−0.928 + 0.371i)36-s + (0.786 + 1.36i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.292 + 0.956i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (677, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.292 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.279859692\)
\(L(\frac12)\) \(\approx\) \(1.279859692\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.235 + 0.971i)T \)
7 \( 1 + (-0.841 + 0.540i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
good2 \( 1 + (-0.995 - 0.0950i)T^{2} \)
5 \( 1 + (0.786 - 0.618i)T^{2} \)
11 \( 1 + (-0.786 + 0.618i)T^{2} \)
13 \( 1 + (1.56 + 0.625i)T + (0.723 + 0.690i)T^{2} \)
17 \( 1 + (-0.654 - 0.755i)T^{2} \)
19 \( 1 + (-0.890 + 1.72i)T + (-0.580 - 0.814i)T^{2} \)
23 \( 1 + (0.888 - 0.458i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-1.02 - 0.809i)T + (0.235 + 0.971i)T^{2} \)
37 \( 1 + (-0.786 - 1.36i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.981 + 0.189i)T^{2} \)
43 \( 1 + (1.80 + 0.822i)T + (0.654 + 0.755i)T^{2} \)
47 \( 1 + (0.841 + 0.540i)T^{2} \)
53 \( 1 + (-0.327 - 0.945i)T^{2} \)
59 \( 1 + (0.723 - 0.690i)T^{2} \)
61 \( 1 + (0.154 - 0.445i)T + (-0.786 - 0.618i)T^{2} \)
71 \( 1 + (0.327 + 0.945i)T^{2} \)
73 \( 1 + (0.286 - 0.247i)T + (0.142 - 0.989i)T^{2} \)
79 \( 1 + (-1.22 + 0.175i)T + (0.959 - 0.281i)T^{2} \)
83 \( 1 + (-0.786 + 0.618i)T^{2} \)
89 \( 1 + (-0.888 - 0.458i)T^{2} \)
97 \( 1 + (0.723 - 1.25i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.756497396175178352318902704638, −8.426214926161133763154917726371, −7.74557859962698195599592981784, −7.15670735821360979080856306718, −6.65501222691167702499265789721, −5.40285386651496853220842321275, −4.83888016643018315580233628207, −3.10256611443265130839153074024, −2.34067445151976027315774684495, −1.16153499467736481530000822308, 1.87306280060931463221214693623, 2.80150556761373211222022391418, 4.01175946194131646164760972612, 4.97434428836328218681500188296, 5.70636750794196212518620297513, 6.46882919999302404786556048637, 7.71052225397825100016054244003, 8.108511193802027840949257332466, 9.462233064460166753953103173769, 9.871890132023637831040204185776

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