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Properties

Label 2-1412-1412.319-c0-0-0
Degree $2$
Conductor $1412$
Sign $-0.987 + 0.157i$
Analytic cond. $0.704679$
Root an. cond. $0.839452$
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  + (−0.415 + 0.909i)2-s + (−0.654 − 0.755i)4-s + (−0.148 + 0.398i)5-s + (0.959 − 0.281i)8-s + (−0.755 − 0.654i)9-s + (−0.300 − 0.300i)10-s + (−1.50 + 1.12i)13-s + (−0.142 + 0.989i)16-s + (−1.25 + 0.368i)17-s + (0.909 − 0.415i)18-s + (0.398 − 0.148i)20-s + (0.619 + 0.536i)25-s + (−0.398 − 1.83i)26-s + (−1.29 + 1.49i)29-s + (−0.841 − 0.540i)32-s + ⋯
L(s)  = 1  + (−0.415 + 0.909i)2-s + (−0.654 − 0.755i)4-s + (−0.148 + 0.398i)5-s + (0.959 − 0.281i)8-s + (−0.755 − 0.654i)9-s + (−0.300 − 0.300i)10-s + (−1.50 + 1.12i)13-s + (−0.142 + 0.989i)16-s + (−1.25 + 0.368i)17-s + (0.909 − 0.415i)18-s + (0.398 − 0.148i)20-s + (0.619 + 0.536i)25-s + (−0.398 − 1.83i)26-s + (−1.29 + 1.49i)29-s + (−0.841 − 0.540i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1412\)    =    \(2^{2} \cdot 353\)
Sign: $-0.987 + 0.157i$
Analytic conductor: \(0.704679\)
Root analytic conductor: \(0.839452\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1412} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1412,\ (\ :0),\ -0.987 + 0.157i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3235595449\)
\(L(\frac12)\) \(\approx\) \(0.3235595449\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 - 0.909i)T \)
353 \( 1 + (0.959 + 0.281i)T \)
good3 \( 1 + (0.755 + 0.654i)T^{2} \)
5 \( 1 + (0.148 - 0.398i)T + (-0.755 - 0.654i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (-0.415 + 0.909i)T^{2} \)
13 \( 1 + (1.50 - 1.12i)T + (0.281 - 0.959i)T^{2} \)
17 \( 1 + (1.25 - 0.368i)T + (0.841 - 0.540i)T^{2} \)
19 \( 1 + (-0.959 + 0.281i)T^{2} \)
23 \( 1 + (0.415 + 0.909i)T^{2} \)
29 \( 1 + (1.29 - 1.49i)T + (-0.142 - 0.989i)T^{2} \)
31 \( 1 + (-0.909 - 0.415i)T^{2} \)
37 \( 1 + (0.459 - 0.841i)T + (-0.540 - 0.841i)T^{2} \)
41 \( 1 + (-1.03 - 1.61i)T + (-0.415 + 0.909i)T^{2} \)
43 \( 1 + (0.415 - 0.909i)T^{2} \)
47 \( 1 + (-0.142 + 0.989i)T^{2} \)
53 \( 1 + (1.86 + 0.697i)T + (0.755 + 0.654i)T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-0.281 + 1.95i)T + (-0.959 - 0.281i)T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + (-0.989 - 0.142i)T^{2} \)
73 \( 1 + (-0.142 + 0.989i)T^{2} \)
79 \( 1 + (0.755 - 0.654i)T^{2} \)
83 \( 1 + (-0.841 - 0.540i)T^{2} \)
89 \( 1 + (0.203 + 0.373i)T + (-0.540 + 0.841i)T^{2} \)
97 \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \)
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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

−9.789166567297754944695734601580, −9.275871097609523774487136789306, −8.622491416493304114017013621105, −7.65613899804644704528766410711, −6.78733481012329610880899713161, −6.47023680238532564037512655602, −5.23465388637584356712797092895, −4.53795491804062519526224396532, −3.31205628840026402262137008647, −1.89539910376946067184207942033, 0.28586819622339563061277034334, 2.22255141606517467599373799573, 2.79098049303846214648744796238, 4.17033320238756636886531596197, 4.92629509223193232375432570850, 5.77759632675625037155094689682, 7.35533025278360433935109673420, 7.78849452576347227609791343604, 8.736543778927465724291349706080, 9.300978953313632937560514099489

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