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Properties

Label 2-1412-1412.1395-c0-0-0
Degree $2$
Conductor $1412$
Sign $-0.397 - 0.917i$
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  + (−0.755 − 0.654i)2-s + (0.142 + 0.989i)4-s + (−0.894 + 1.78i)5-s + (0.540 − 0.841i)8-s + (0.599 + 0.800i)9-s + (1.84 − 0.764i)10-s + (−0.491 + 0.396i)13-s + (−0.959 + 0.281i)16-s + (−0.153 + 0.239i)17-s + (0.0713 − 0.997i)18-s + (−1.89 − 0.631i)20-s + (−1.79 − 2.39i)25-s + (0.631 + 0.0225i)26-s + (−0.948 − 0.136i)29-s + (0.909 + 0.415i)32-s + ⋯
L(s)  = 1  + (−0.755 − 0.654i)2-s + (0.142 + 0.989i)4-s + (−0.894 + 1.78i)5-s + (0.540 − 0.841i)8-s + (0.599 + 0.800i)9-s + (1.84 − 0.764i)10-s + (−0.491 + 0.396i)13-s + (−0.959 + 0.281i)16-s + (−0.153 + 0.239i)17-s + (0.0713 − 0.997i)18-s + (−1.89 − 0.631i)20-s + (−1.79 − 2.39i)25-s + (0.631 + 0.0225i)26-s + (−0.948 − 0.136i)29-s + (0.909 + 0.415i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1412\)    =    \(2^{2} \cdot 353\)
Sign: $-0.397 - 0.917i$
Analytic conductor: \(0.704679\)
Root analytic conductor: \(0.839452\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1412} (1395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1412,\ (\ :0),\ -0.397 - 0.917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4801296914\)
\(L(\frac12)\) \(\approx\) \(0.4801296914\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.755 + 0.654i)T \)
353 \( 1 + (-0.841 + 0.540i)T \)
good3 \( 1 + (-0.599 - 0.800i)T^{2} \)
5 \( 1 + (0.894 - 1.78i)T + (-0.599 - 0.800i)T^{2} \)
7 \( 1 + (0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.654 + 0.755i)T^{2} \)
13 \( 1 + (0.491 - 0.396i)T + (0.212 - 0.977i)T^{2} \)
17 \( 1 + (0.153 - 0.239i)T + (-0.415 - 0.909i)T^{2} \)
19 \( 1 + (0.540 - 0.841i)T^{2} \)
23 \( 1 + (-0.755 + 0.654i)T^{2} \)
29 \( 1 + (0.948 + 0.136i)T + (0.959 + 0.281i)T^{2} \)
31 \( 1 + (0.0713 + 0.997i)T^{2} \)
37 \( 1 + (-0.936 + 0.650i)T + (0.349 - 0.936i)T^{2} \)
41 \( 1 + (0.587 - 1.57i)T + (-0.755 - 0.654i)T^{2} \)
43 \( 1 + (0.755 + 0.654i)T^{2} \)
47 \( 1 + (-0.281 - 0.959i)T^{2} \)
53 \( 1 + (1.83 - 0.611i)T + (0.800 - 0.599i)T^{2} \)
59 \( 1 + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (-0.494 - 1.68i)T + (-0.841 + 0.540i)T^{2} \)
67 \( 1 + (-0.707 + 0.707i)T^{2} \)
71 \( 1 + (0.877 + 0.479i)T^{2} \)
73 \( 1 + (-1.39 + 0.201i)T + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (0.599 - 0.800i)T^{2} \)
83 \( 1 + (0.415 - 0.909i)T^{2} \)
89 \( 1 + (-0.0407 + 0.0586i)T + (-0.349 - 0.936i)T^{2} \)
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

−10.10144898289856645368120040322, −9.489437213065308078231061196744, −8.174551855518805528345528437542, −7.66281072400490480806133249396, −7.06504408374976022606764283145, −6.31056046408085742468358902549, −4.57695459667218947921254646340, −3.76229131267344709315991770175, −2.84442942217698531202105830855, −1.98154227183061097305154639846, 0.51135723097865521533394833434, 1.67301761975494594902695597599, 3.66351100335593123092904784909, 4.69146184336595053289802002989, 5.23706091691604454105485892598, 6.32195090300310280751979261142, 7.36461529380963425353052664106, 7.931108374179104323680935933443, 8.665426487000232883885727980575, 9.400943291450861519173907643944

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