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Properties

Label 2-1412-1412.1339-c0-0-0
Degree $2$
Conductor $1412$
Sign $-0.201 - 0.979i$
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  + (0.540 + 0.841i)2-s + (−0.415 + 0.909i)4-s + (1.95 + 0.353i)5-s + (−0.989 + 0.142i)8-s + (−0.936 − 0.349i)9-s + (0.760 + 1.83i)10-s + (−0.735 + 1.46i)13-s + (−0.654 − 0.755i)16-s + (−0.822 + 0.118i)17-s + (−0.212 − 0.977i)18-s + (−1.13 + 1.63i)20-s + (2.76 + 1.03i)25-s + (−1.63 + 0.175i)26-s + (1.81 − 0.828i)29-s + (0.281 − 0.959i)32-s + ⋯
L(s)  = 1  + (0.540 + 0.841i)2-s + (−0.415 + 0.909i)4-s + (1.95 + 0.353i)5-s + (−0.989 + 0.142i)8-s + (−0.936 − 0.349i)9-s + (0.760 + 1.83i)10-s + (−0.735 + 1.46i)13-s + (−0.654 − 0.755i)16-s + (−0.822 + 0.118i)17-s + (−0.212 − 0.977i)18-s + (−1.13 + 1.63i)20-s + (2.76 + 1.03i)25-s + (−1.63 + 0.175i)26-s + (1.81 − 0.828i)29-s + (0.281 − 0.959i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1412\)    =    \(2^{2} \cdot 353\)
Sign: $-0.201 - 0.979i$
Analytic conductor: \(0.704679\)
Root analytic conductor: \(0.839452\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1412} (1339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1412,\ (\ :0),\ -0.201 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.662048532\)
\(L(\frac12)\) \(\approx\) \(1.662048532\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.540 - 0.841i)T \)
353 \( 1 + (0.142 - 0.989i)T \)
good3 \( 1 + (0.936 + 0.349i)T^{2} \)
5 \( 1 + (-1.95 - 0.353i)T + (0.936 + 0.349i)T^{2} \)
7 \( 1 + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.841 - 0.540i)T^{2} \)
13 \( 1 + (0.735 - 1.46i)T + (-0.599 - 0.800i)T^{2} \)
17 \( 1 + (0.822 - 0.118i)T + (0.959 - 0.281i)T^{2} \)
19 \( 1 + (-0.989 + 0.142i)T^{2} \)
23 \( 1 + (0.540 - 0.841i)T^{2} \)
29 \( 1 + (-1.81 + 0.828i)T + (0.654 - 0.755i)T^{2} \)
31 \( 1 + (-0.212 + 0.977i)T^{2} \)
37 \( 1 + (-0.479 + 1.87i)T + (-0.877 - 0.479i)T^{2} \)
41 \( 1 + (0.249 + 0.136i)T + (0.540 + 0.841i)T^{2} \)
43 \( 1 + (-0.540 - 0.841i)T^{2} \)
47 \( 1 + (0.755 - 0.654i)T^{2} \)
53 \( 1 + (0.834 + 1.20i)T + (-0.349 + 0.936i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.107 - 0.0934i)T + (0.142 - 0.989i)T^{2} \)
67 \( 1 + (0.707 - 0.707i)T^{2} \)
71 \( 1 + (0.0713 - 0.997i)T^{2} \)
73 \( 1 + (1.28 + 0.587i)T + (0.654 + 0.755i)T^{2} \)
79 \( 1 + (-0.936 + 0.349i)T^{2} \)
83 \( 1 + (-0.959 - 0.281i)T^{2} \)
89 \( 1 + (-0.207 + 0.0528i)T + (0.877 - 0.479i)T^{2} \)
97 \( 1 + (-0.239 + 1.66i)T + (-0.959 - 0.281i)T^{2} \)
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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

−9.615620145413746365425020652388, −9.184280903529025354570587121575, −8.507695235818945745694065513412, −7.17946573637779019861104660101, −6.46199482001379471455088864071, −6.04989295541105004155545108948, −5.17374399283685302081239555212, −4.30123418227556323178226726776, −2.82813296647257273409622408604, −2.15206688753102560393290719859, 1.27332856907842522633015826081, 2.59794112185846907187991205470, 2.86399092100264983833415320337, 4.77802765285175138182878131196, 5.18784105677734693899982683055, 5.97216101274212258942116381191, 6.64259662654711432861741365546, 8.289742592933928355013488507900, 8.931397687633661899996678756351, 9.769872815061661981085290376689

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