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Properties

Label 2-1407-1407.284-c0-0-0
Degree $2$
Conductor $1407$
Sign $-0.262 - 0.964i$
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  + (−0.995 − 0.0950i)3-s + (−0.786 + 0.618i)4-s + (−0.654 − 0.755i)7-s + (0.981 + 0.189i)9-s + (0.841 − 0.540i)12-s + (1.16 + 0.600i)13-s + (0.235 − 0.971i)16-s + (−1.95 + 0.376i)19-s + (0.580 + 0.814i)21-s + (0.0475 + 0.998i)25-s + (−0.959 − 0.281i)27-s + (0.981 + 0.189i)28-s + (−0.0913 + 1.91i)31-s + (−0.888 + 0.458i)36-s + (−0.0475 + 0.0824i)37-s + ⋯
L(s)  = 1  + (−0.995 − 0.0950i)3-s + (−0.786 + 0.618i)4-s + (−0.654 − 0.755i)7-s + (0.981 + 0.189i)9-s + (0.841 − 0.540i)12-s + (1.16 + 0.600i)13-s + (0.235 − 0.971i)16-s + (−1.95 + 0.376i)19-s + (0.580 + 0.814i)21-s + (0.0475 + 0.998i)25-s + (−0.959 − 0.281i)27-s + (0.981 + 0.189i)28-s + (−0.0913 + 1.91i)31-s + (−0.888 + 0.458i)36-s + (−0.0475 + 0.0824i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $-0.262 - 0.964i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (284, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ -0.262 - 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4050917819\)
\(L(\frac12)\) \(\approx\) \(0.4050917819\)
\(L(1)\) 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 \( 1 + (0.654 + 0.755i)T \)
67 \( 1 + (-0.841 + 0.540i)T \)
good2 \( 1 + (0.786 - 0.618i)T^{2} \)
5 \( 1 + (-0.0475 - 0.998i)T^{2} \)
11 \( 1 + (-0.0475 - 0.998i)T^{2} \)
13 \( 1 + (-1.16 - 0.600i)T + (0.580 + 0.814i)T^{2} \)
17 \( 1 + (0.959 + 0.281i)T^{2} \)
19 \( 1 + (1.95 - 0.376i)T + (0.928 - 0.371i)T^{2} \)
23 \( 1 + (-0.981 - 0.189i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.0913 - 1.91i)T + (-0.995 - 0.0950i)T^{2} \)
37 \( 1 + (0.0475 - 0.0824i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.235 - 0.971i)T^{2} \)
43 \( 1 + (0.239 - 1.66i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 + (0.654 - 0.755i)T^{2} \)
53 \( 1 + (-0.723 - 0.690i)T^{2} \)
59 \( 1 + (-0.580 + 0.814i)T^{2} \)
61 \( 1 + (1.44 - 1.37i)T + (0.0475 - 0.998i)T^{2} \)
71 \( 1 + (-0.723 - 0.690i)T^{2} \)
73 \( 1 + (0.452 + 0.132i)T + (0.841 + 0.540i)T^{2} \)
79 \( 1 + (-0.0800 + 0.0514i)T + (0.415 - 0.909i)T^{2} \)
83 \( 1 + (-0.0475 - 0.998i)T^{2} \)
89 \( 1 + (-0.981 + 0.189i)T^{2} \)
97 \( 1 + (0.580 + 1.00i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.10652693646715820514055906856, −9.148433472698079478444741845975, −8.460076455474593091543274248605, −7.44772661763951698207237508027, −6.64662370586002167687095341281, −6.01831841289597306761813243062, −4.79034961480429476747752393584, −4.13556743699902679527470934985, −3.32860772016070323714108736529, −1.39377915662488730271529572060, 0.41644615414517080884242829857, 2.09058092037446007294045424891, 3.75636197841131192957958657339, 4.50129537932064475485970851058, 5.54678951609816069466720338991, 6.09965889155138279007419423395, 6.62973882357196597173829240236, 8.122804381003099647493704104291, 8.832492159674091321707713166573, 9.582950557088135900938804764436

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