Properties

Label 2-392-56.51-c2-0-11
Degree $2$
Conductor $392$
Sign $-0.286 - 0.958i$
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  + (−1.97 − 0.323i)2-s + (1.70 + 2.95i)3-s + (3.79 + 1.27i)4-s + (−1.34 − 0.774i)5-s + (−2.41 − 6.38i)6-s + (−7.07 − 3.74i)8-s + (−1.32 + 2.30i)9-s + (2.39 + 1.96i)10-s + (2.24 + 3.88i)11-s + (2.70 + 13.3i)12-s − 1.54i·13-s − 5.29i·15-s + (12.7 + 9.66i)16-s + (11.8 + 20.4i)17-s + (3.36 − 4.11i)18-s + (−12.4 + 21.5i)19-s + ⋯
L(s)  = 1  + (−0.986 − 0.161i)2-s + (0.569 + 0.985i)3-s + (0.947 + 0.318i)4-s + (−0.268 − 0.154i)5-s + (−0.402 − 1.06i)6-s + (−0.883 − 0.467i)8-s + (−0.147 + 0.255i)9-s + (0.239 + 0.196i)10-s + (0.203 + 0.353i)11-s + (0.225 + 1.11i)12-s − 0.119i·13-s − 0.352i·15-s + (0.796 + 0.604i)16-s + (0.695 + 1.20i)17-s + (0.186 − 0.228i)18-s + (−0.654 + 1.13i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.286 - 0.958i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ -0.286 - 0.958i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.662521 + 0.889471i\)
\(L(\frac12)\) \(\approx\) \(0.662521 + 0.889471i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.97 + 0.323i)T \)
7 \( 1 \)
good3 \( 1 + (-1.70 - 2.95i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (1.34 + 0.774i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-2.24 - 3.88i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 1.54iT - 169T^{2} \)
17 \( 1 + (-11.8 - 20.4i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (12.4 - 21.5i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-30.5 - 17.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 22.4iT - 841T^{2} \)
31 \( 1 + (40.4 - 23.3i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (50.7 + 29.2i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 26.9T + 1.68e3T^{2} \)
43 \( 1 + 17.1T + 1.84e3T^{2} \)
47 \( 1 + (-31.2 - 18.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (84.7 - 48.9i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-30.7 - 53.3i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-32.6 - 18.8i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-16.6 - 28.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 102. iT - 5.04e3T^{2} \)
73 \( 1 + (-34.6 - 60.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-33.5 - 19.3i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 3.61T + 6.88e3T^{2} \)
89 \( 1 + (-22.0 + 38.1i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 96.1T + 9.40e3T^{2} \)
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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

−10.83447007182672324360570191051, −10.41532597869834545521051741755, −9.459423343375504078315364045526, −8.795100783867377791589373163062, −7.982296953528682148722962637753, −6.93464231234156808592954282017, −5.62312801357325811393005443804, −4.03418178907144871840195362107, −3.26528809515673103005807961746, −1.58105303880056688812369512005, 0.64136042506000896168684520372, 2.09122054490692145395137687905, 3.17179893582484471589361714285, 5.17018245017311139803484951041, 6.63785586731138234082859146871, 7.17862354294477861587133177623, 8.003938288553907538955542276501, 8.857752922443501698914231436222, 9.629595551408923608122442292643, 10.91110450249774552189886355469

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