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Properties

Label 2-392-392.253-c1-0-14
Degree $2$
Conductor $392$
Sign $-0.762 - 0.646i$
Analytic cond. $3.13013$
Root an. cond. $1.76921$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.961 + 1.03i)2-s + (0.947 + 1.96i)3-s + (−0.150 + 1.99i)4-s + (−0.764 − 1.58i)5-s + (−1.12 + 2.87i)6-s + (−0.789 + 2.52i)7-s + (−2.21 + 1.76i)8-s + (−1.10 + 1.38i)9-s + (0.910 − 2.31i)10-s + (3.11 − 2.48i)11-s + (−4.06 + 1.59i)12-s + (−2.68 + 2.14i)13-s + (−3.37 + 1.60i)14-s + (2.39 − 3.00i)15-s + (−3.95 − 0.602i)16-s + (1.58 + 6.93i)17-s + ⋯
L(s)  = 1  + (0.679 + 0.733i)2-s + (0.547 + 1.13i)3-s + (−0.0754 + 0.997i)4-s + (−0.341 − 0.709i)5-s + (−0.461 + 1.17i)6-s + (−0.298 + 0.954i)7-s + (−0.782 + 0.622i)8-s + (−0.367 + 0.460i)9-s + (0.288 − 0.733i)10-s + (0.938 − 0.748i)11-s + (−1.17 + 0.459i)12-s + (−0.745 + 0.594i)13-s + (−0.902 + 0.430i)14-s + (0.619 − 0.776i)15-s + (−0.988 − 0.150i)16-s + (0.383 + 1.68i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 - 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.762 - 0.646i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ -0.762 - 0.646i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.706789 + 1.92721i\)
\(L(\frac12)\) \(\approx\) \(0.706789 + 1.92721i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.961 - 1.03i)T \)
7 \( 1 + (0.789 - 2.52i)T \)
good3 \( 1 + (-0.947 - 1.96i)T + (-1.87 + 2.34i)T^{2} \)
5 \( 1 + (0.764 + 1.58i)T + (-3.11 + 3.90i)T^{2} \)
11 \( 1 + (-3.11 + 2.48i)T + (2.44 - 10.7i)T^{2} \)
13 \( 1 + (2.68 - 2.14i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (-1.58 - 6.93i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 + 5.77iT - 19T^{2} \)
23 \( 1 + (-0.268 + 1.17i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-0.805 + 0.183i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 - 3.78T + 31T^{2} \)
37 \( 1 + (0.185 - 0.0422i)T + (33.3 - 16.0i)T^{2} \)
41 \( 1 + (-3.78 + 1.82i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (-2.87 + 5.97i)T + (-26.8 - 33.6i)T^{2} \)
47 \( 1 + (-0.401 - 0.503i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-6.54 - 1.49i)T + (47.7 + 22.9i)T^{2} \)
59 \( 1 + (4.89 - 10.1i)T + (-36.7 - 46.1i)T^{2} \)
61 \( 1 + (10.6 - 2.43i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 - 8.09iT - 67T^{2} \)
71 \( 1 + (-2.28 + 10.0i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-5.30 + 6.65i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 - 8.76T + 79T^{2} \)
83 \( 1 + (-0.0236 - 0.0188i)T + (18.4 + 80.9i)T^{2} \)
89 \( 1 + (9.91 - 12.4i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + 9.31T + 97T^{2} \)
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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

−12.05199050237363902136834385974, −10.77277138270601702003081088977, −9.357263374613836270287479084028, −8.879300073825948729261602216280, −8.265289050124713629976318071510, −6.75504204038323101775367653614, −5.76148404563753058910817889666, −4.62162963467457953594629321272, −3.96814072213212930448995587257, −2.79932524593813786537723339027, 1.16322601829160028984950192001, 2.61418710747507417371358479381, 3.56258330864258600403173130528, 4.81874389699461220753943234630, 6.40548745330548606717796632557, 7.16876523050618488699784564062, 7.77176483102801959922143715303, 9.469111981947419358171131407909, 10.09783412125170373505385318958, 11.15677549839015186156023173770

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