Properties

Label 2-392-56.45-c2-0-4
Degree $2$
Conductor $392$
Sign $0.558 - 0.829i$
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  + (0.687 − 1.87i)2-s + (−1.94 + 3.37i)3-s + (−3.05 − 2.58i)4-s + (−4.42 − 7.67i)5-s + (4.99 + 5.97i)6-s + (−6.95 + 3.95i)8-s + (−3.09 − 5.35i)9-s + (−17.4 + 3.04i)10-s + (3.15 + 1.82i)11-s + (14.6 − 5.27i)12-s − 7.79·13-s + 34.5·15-s + (2.65 + 15.7i)16-s + (9.07 + 5.23i)17-s + (−12.1 + 2.12i)18-s + (5.39 + 9.34i)19-s + ⋯
L(s)  = 1  + (0.343 − 0.939i)2-s + (−0.649 + 1.12i)3-s + (−0.763 − 0.645i)4-s + (−0.885 − 1.53i)5-s + (0.832 + 0.996i)6-s + (−0.868 + 0.494i)8-s + (−0.343 − 0.594i)9-s + (−1.74 + 0.304i)10-s + (0.287 + 0.165i)11-s + (1.22 − 0.439i)12-s − 0.599·13-s + 2.30·15-s + (0.165 + 0.986i)16-s + (0.533 + 0.308i)17-s + (−0.676 + 0.117i)18-s + (0.283 + 0.491i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.558 - 0.829i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ 0.558 - 0.829i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.521415 + 0.277412i\)
\(L(\frac12)\) \(\approx\) \(0.521415 + 0.277412i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.687 + 1.87i)T \)
7 \( 1 \)
good3 \( 1 + (1.94 - 3.37i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (4.42 + 7.67i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (-3.15 - 1.82i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 7.79T + 169T^{2} \)
17 \( 1 + (-9.07 - 5.23i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-5.39 - 9.34i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-6.45 - 11.1i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 17.2iT - 841T^{2} \)
31 \( 1 + (-26.1 - 15.1i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (34.2 - 19.7i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 73.6iT - 1.68e3T^{2} \)
43 \( 1 - 40.8iT - 1.84e3T^{2} \)
47 \( 1 + (-36.2 + 20.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-5.55 - 3.20i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (7.95 - 13.7i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (6.07 + 10.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (6.75 + 3.89i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 41.3T + 5.04e3T^{2} \)
73 \( 1 + (77.6 + 44.8i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (35.3 + 61.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 60.8T + 6.88e3T^{2} \)
89 \( 1 + (-23.4 + 13.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 3.26iT - 9.40e3T^{2} \)
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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

−11.51445892758336753606244737567, −10.29377484251544679065844168344, −9.681076178919255458110867391773, −8.828312775346747744313239469045, −7.83243646377700642598013231535, −5.84285595935436811310099192360, −4.88037518717361996612604235388, −4.46333139093990521867738714506, −3.44238549088222476404675762496, −1.24301891466296114909958306940, 0.28811494425308392887346627471, 2.78503287376925912603570508373, 3.97429506582686679926885342054, 5.47519016373677832728864379156, 6.47814075827078118256131514702, 7.23761093657963137023805532031, 7.43887518635706738481986216025, 8.728128919781682433766151804997, 10.16248846672417192433932654886, 11.26270561026632056061869516862

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