Properties

Label 2-1412-1412.191-c0-0-0
Degree $2$
Conductor $1412$
Sign $0.245 - 0.969i$
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  + (0.654 + 0.755i)2-s + (−0.142 + 0.989i)4-s + (0.697 − 0.0498i)5-s + (−0.841 + 0.540i)8-s + (0.989 − 0.142i)9-s + (0.494 + 0.494i)10-s + (−0.0683 − 0.125i)13-s + (−0.959 − 0.281i)16-s + (0.239 − 0.153i)17-s + (0.755 + 0.654i)18-s + (−0.0498 + 0.697i)20-s + (−0.506 + 0.0727i)25-s + (0.0498 − 0.133i)26-s + (0.0801 + 0.557i)29-s + (−0.415 − 0.909i)32-s + ⋯
L(s)  = 1  + (0.654 + 0.755i)2-s + (−0.142 + 0.989i)4-s + (0.697 − 0.0498i)5-s + (−0.841 + 0.540i)8-s + (0.989 − 0.142i)9-s + (0.494 + 0.494i)10-s + (−0.0683 − 0.125i)13-s + (−0.959 − 0.281i)16-s + (0.239 − 0.153i)17-s + (0.755 + 0.654i)18-s + (−0.0498 + 0.697i)20-s + (−0.506 + 0.0727i)25-s + (0.0498 − 0.133i)26-s + (0.0801 + 0.557i)29-s + (−0.415 − 0.909i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1412\)    =    \(2^{2} \cdot 353\)
Sign: $0.245 - 0.969i$
Analytic conductor: \(0.704679\)
Root analytic conductor: \(0.839452\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1412} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1412,\ (\ :0),\ 0.245 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.716208148\)
\(L(\frac12)\) \(\approx\) \(1.716208148\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.654 - 0.755i)T \)
353 \( 1 + (-0.841 - 0.540i)T \)
good3 \( 1 + (-0.989 + 0.142i)T^{2} \)
5 \( 1 + (-0.697 + 0.0498i)T + (0.989 - 0.142i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (0.654 + 0.755i)T^{2} \)
13 \( 1 + (0.0683 + 0.125i)T + (-0.540 + 0.841i)T^{2} \)
17 \( 1 + (-0.239 + 0.153i)T + (0.415 - 0.909i)T^{2} \)
19 \( 1 + (0.841 - 0.540i)T^{2} \)
23 \( 1 + (-0.654 + 0.755i)T^{2} \)
29 \( 1 + (-0.0801 - 0.557i)T + (-0.959 + 0.281i)T^{2} \)
31 \( 1 + (-0.755 + 0.654i)T^{2} \)
37 \( 1 + (0.0903 - 0.415i)T + (-0.909 - 0.415i)T^{2} \)
41 \( 1 + (1.53 + 0.698i)T + (0.654 + 0.755i)T^{2} \)
43 \( 1 + (-0.654 - 0.755i)T^{2} \)
47 \( 1 + (-0.959 - 0.281i)T^{2} \)
53 \( 1 + (-0.0855 - 1.19i)T + (-0.989 + 0.142i)T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.540 + 0.158i)T + (0.841 + 0.540i)T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + (0.281 - 0.959i)T^{2} \)
73 \( 1 + (-0.959 - 0.281i)T^{2} \)
79 \( 1 + (-0.989 - 0.142i)T^{2} \)
83 \( 1 + (-0.415 - 0.909i)T^{2} \)
89 \( 1 + (0.148 + 0.682i)T + (-0.909 + 0.415i)T^{2} \)
97 \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \)
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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

−9.814199759689515828080208654489, −9.048515649487016814491676884345, −8.164133508596826849940656455589, −7.26666467067336139779210705796, −6.67011952648189075412368421853, −5.75828990523155719114623240239, −5.02552426303545266426350160696, −4.10936548859201459521017795683, −3.12489593977970240901648269693, −1.80129263157510118824719142101, 1.44154676304798174163277548108, 2.32698867437060913901012158724, 3.53348917786873847221825450377, 4.43848372479185458932825249525, 5.27120618788128573045444547080, 6.15418890429800418278868337073, 6.88957483011696029741459472693, 7.985438795035084867652861518259, 9.135669740325867967174003333701, 9.863380854322864510333547890105

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