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Properties

Label 2-392-8.3-c2-0-30
Degree $2$
Conductor $392$
Sign $-0.163 + 0.986i$
Analytic cond. $10.6812$
Root an. cond. $3.26821$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.67 − 1.09i)2-s − 4.56·3-s + (1.60 + 3.66i)4-s − 5.73i·5-s + (7.64 + 4.99i)6-s + (1.30 − 7.89i)8-s + 11.8·9-s + (−6.26 + 9.60i)10-s − 1.40·11-s + (−7.34 − 16.7i)12-s + 19.0i·13-s + 26.1i·15-s + (−10.8 + 11.7i)16-s + 32.2·17-s + (−19.8 − 12.9i)18-s − 12.5·19-s + ⋯
L(s)  = 1  + (−0.837 − 0.546i)2-s − 1.52·3-s + (0.402 + 0.915i)4-s − 1.14i·5-s + (1.27 + 0.832i)6-s + (0.163 − 0.986i)8-s + 1.31·9-s + (−0.626 + 0.960i)10-s − 0.127·11-s + (−0.612 − 1.39i)12-s + 1.46i·13-s + 1.74i·15-s + (−0.676 + 0.736i)16-s + 1.89·17-s + (−1.10 − 0.720i)18-s − 0.661·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.163 + 0.986i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ -0.163 + 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.354854 - 0.418579i\)
\(L(\frac12)\) \(\approx\) \(0.354854 - 0.418579i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.67 + 1.09i)T \)
7 \( 1 \)
good3 \( 1 + 4.56T + 9T^{2} \)
5 \( 1 + 5.73iT - 25T^{2} \)
11 \( 1 + 1.40T + 121T^{2} \)
13 \( 1 - 19.0iT - 169T^{2} \)
17 \( 1 - 32.2T + 289T^{2} \)
19 \( 1 + 12.5T + 361T^{2} \)
23 \( 1 - 15.8iT - 529T^{2} \)
29 \( 1 + 3.29iT - 841T^{2} \)
31 \( 1 + 22.6iT - 961T^{2} \)
37 \( 1 + 54.1iT - 1.36e3T^{2} \)
41 \( 1 - 7.59T + 1.68e3T^{2} \)
43 \( 1 + 20.8T + 1.84e3T^{2} \)
47 \( 1 + 21.6iT - 2.20e3T^{2} \)
53 \( 1 - 0.356iT - 2.80e3T^{2} \)
59 \( 1 + 26.8T + 3.48e3T^{2} \)
61 \( 1 + 86.2iT - 3.72e3T^{2} \)
67 \( 1 - 114.T + 4.48e3T^{2} \)
71 \( 1 + 104. iT - 5.04e3T^{2} \)
73 \( 1 - 24.3T + 5.32e3T^{2} \)
79 \( 1 - 117. iT - 6.24e3T^{2} \)
83 \( 1 + 79.2T + 6.88e3T^{2} \)
89 \( 1 + 2.66T + 7.92e3T^{2} \)
97 \( 1 - 52.0T + 9.40e3T^{2} \)
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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

−11.00620220230852533159754364413, −9.929884076072802676590850790266, −9.271118584366143896605364467148, −8.174650353943457375603894058143, −7.12045765327173756478935096583, −6.01527561855760897252370938635, −4.99029054396465010724801619217, −3.89768946721223058537500190063, −1.69779087256037813366955644484, −0.54293764238753195314848613684, 0.940363236563343918345989603544, 2.99138919944400368940478642214, 5.04167481522672441297585431592, 5.83501341122100655828275886419, 6.56022142034844085542365150646, 7.43787266314749762946369764637, 8.354020166447226972368487548138, 10.13218921125354346143976252755, 10.26354220688807215336240834031, 11.03159438666655994010518915399

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