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Properties

Label 2-392-56.51-c2-0-40
Degree $2$
Conductor $392$
Sign $-0.0436 - 0.999i$
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  + (1.98 + 0.234i)2-s + (2.66 + 4.61i)3-s + (3.88 + 0.933i)4-s + (1.86 + 1.07i)5-s + (4.21 + 9.79i)6-s + (7.50 + 2.76i)8-s + (−9.71 + 16.8i)9-s + (3.45 + 2.58i)10-s + (−2.62 − 4.55i)11-s + (6.06 + 20.4i)12-s − 21.4i·13-s + 11.5i·15-s + (14.2 + 7.26i)16-s + (0.463 + 0.802i)17-s + (−23.2 + 31.1i)18-s + (−2.96 + 5.13i)19-s + ⋯
L(s)  = 1  + (0.993 + 0.117i)2-s + (0.888 + 1.53i)3-s + (0.972 + 0.233i)4-s + (0.373 + 0.215i)5-s + (0.701 + 1.63i)6-s + (0.938 + 0.345i)8-s + (−1.07 + 1.86i)9-s + (0.345 + 0.258i)10-s + (−0.239 − 0.414i)11-s + (0.505 + 1.70i)12-s − 1.64i·13-s + 0.766i·15-s + (0.891 + 0.453i)16-s + (0.0272 + 0.0472i)17-s + (−1.29 + 1.73i)18-s + (−0.156 + 0.270i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0436 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0436 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.0436 - 0.999i$
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: \((2,\ 392,\ (\ :1),\ -0.0436 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.99387 + 3.12738i\)
\(L(\frac12)\) \(\approx\) \(2.99387 + 3.12738i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.98 - 0.234i)T \)
7 \( 1 \)
good3 \( 1 + (-2.66 - 4.61i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-1.86 - 1.07i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (2.62 + 4.55i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 21.4iT - 169T^{2} \)
17 \( 1 + (-0.463 - 0.802i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (2.96 - 5.13i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (7.52 + 4.34i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 9.42iT - 841T^{2} \)
31 \( 1 + (-29.8 + 17.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-11.0 - 6.40i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 43.1T + 1.68e3T^{2} \)
43 \( 1 + 41.7T + 1.84e3T^{2} \)
47 \( 1 + (-39.8 - 22.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (64.5 - 37.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-26.8 - 46.4i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (24.0 + 13.9i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-39.2 - 67.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 74.5iT - 5.04e3T^{2} \)
73 \( 1 + (-16.8 - 29.1i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-26.1 - 15.1i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 72.9T + 6.88e3T^{2} \)
89 \( 1 + (27.4 - 47.4i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 53.7T + 9.40e3T^{2} \)
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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

−11.09812574498552045902517895677, −10.34031421053958744491853158365, −9.863338644488164736263678462976, −8.399508655857985389943779255709, −7.86220021593259466766734306816, −6.15474419119509281527044749518, −5.29866452251523509659139453134, −4.30829939099184657194733229411, −3.28729843869003149060095528458, −2.53201058509185907802973865612, 1.54302782920484941221545471472, 2.26173099050145286406535832937, 3.55669884443040275769153141062, 4.96100574375445145901780897621, 6.35879928049296249903068147626, 6.88510531339981580316676401754, 7.79214179725776480625561142984, 8.863361049320925468229729153664, 9.873964136918489496022536996933, 11.37806739129191763863895474590

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