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Properties

Label 2-1407-1407.530-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.976 + 0.215i$
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  + (−0.995 + 0.0950i)3-s + (0.142 − 0.989i)4-s + (0.888 + 0.458i)7-s + (0.981 − 0.189i)9-s + (−0.0475 + 0.998i)12-s + (−0.0623 + 1.30i)13-s + (−0.959 − 0.281i)16-s + (1.23 + 1.06i)19-s + (−0.928 − 0.371i)21-s + (0.0475 − 0.998i)25-s + (−0.959 + 0.281i)27-s + (0.580 − 0.814i)28-s + (1.61 − 1.03i)31-s + (−0.0475 − 0.998i)36-s − 1.77·37-s + ⋯
L(s)  = 1  + (−0.995 + 0.0950i)3-s + (0.142 − 0.989i)4-s + (0.888 + 0.458i)7-s + (0.981 − 0.189i)9-s + (−0.0475 + 0.998i)12-s + (−0.0623 + 1.30i)13-s + (−0.959 − 0.281i)16-s + (1.23 + 1.06i)19-s + (−0.928 − 0.371i)21-s + (0.0475 − 0.998i)25-s + (−0.959 + 0.281i)27-s + (0.580 − 0.814i)28-s + (1.61 − 1.03i)31-s + (−0.0475 − 0.998i)36-s − 1.77·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.976 + 0.215i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (530, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.976 + 0.215i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9348888397\)
\(L(\frac12)\) \(\approx\) \(0.9348888397\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.995 - 0.0950i)T \)
7 \( 1 + (-0.888 - 0.458i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
good2 \( 1 + (-0.142 + 0.989i)T^{2} \)
5 \( 1 + (-0.0475 + 0.998i)T^{2} \)
11 \( 1 + (0.841 + 0.540i)T^{2} \)
13 \( 1 + (0.0623 - 1.30i)T + (-0.995 - 0.0950i)T^{2} \)
17 \( 1 + (0.723 + 0.690i)T^{2} \)
19 \( 1 + (-1.23 - 1.06i)T + (0.142 + 0.989i)T^{2} \)
23 \( 1 + (0.654 + 0.755i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-1.61 + 1.03i)T + (0.415 - 0.909i)T^{2} \)
37 \( 1 + 1.77T + T^{2} \)
41 \( 1 + (-0.723 - 0.690i)T^{2} \)
43 \( 1 + (-1.07 + 0.153i)T + (0.959 - 0.281i)T^{2} \)
47 \( 1 + (0.981 - 0.189i)T^{2} \)
53 \( 1 + (0.723 - 0.690i)T^{2} \)
59 \( 1 + (0.580 + 0.814i)T^{2} \)
61 \( 1 + (-1.11 + 0.326i)T + (0.841 - 0.540i)T^{2} \)
71 \( 1 + (-0.235 - 0.971i)T^{2} \)
73 \( 1 + (1.34 + 0.325i)T + (0.888 + 0.458i)T^{2} \)
79 \( 1 + (-0.915 - 0.0436i)T + (0.995 + 0.0950i)T^{2} \)
83 \( 1 + (-0.888 + 0.458i)T^{2} \)
89 \( 1 + (0.981 + 0.189i)T^{2} \)
97 \( 1 + (0.995 + 1.72i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.877427340311027580721470067551, −9.155687929312549963379622424212, −8.077615611502508792686337154122, −7.06036675896665412191407822720, −6.28928763954075354710385063488, −5.57698018161624639865901173799, −4.86544891839337587541412512937, −4.08810090059521490753317329939, −2.19836728086716354875037631179, −1.23612880282652823563804675053, 1.15818963831590445725301818884, 2.77001055210288302093718470556, 3.83002994119829911433899739320, 4.92831460065955337334053306956, 5.39405065197295140349572889399, 6.75420584007319101700981025265, 7.32165910779540762159081840992, 7.973547150463845833246162385212, 8.832410669114951718261136509997, 9.986034166314865369098922297540

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