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Properties

Label 2-392-56.51-c2-0-41
Degree $2$
Conductor $392$
Sign $0.818 - 0.575i$
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  + (1.78 − 0.903i)2-s + (2.28 + 3.95i)3-s + (2.36 − 3.22i)4-s + (4.96 + 2.86i)5-s + (7.64 + 4.99i)6-s + (1.30 − 7.89i)8-s + (−5.93 + 10.2i)9-s + (11.4 + 0.628i)10-s + (0.700 + 1.21i)11-s + (18.1 + 1.99i)12-s + 19.0i·13-s + 26.1i·15-s + (−4.79 − 15.2i)16-s + (−16.1 − 27.9i)17-s + (−1.29 + 23.6i)18-s + (6.28 − 10.8i)19-s + ⋯
L(s)  = 1  + (0.892 − 0.451i)2-s + (0.761 + 1.31i)3-s + (0.591 − 0.806i)4-s + (0.992 + 0.573i)5-s + (1.27 + 0.832i)6-s + (0.163 − 0.986i)8-s + (−0.658 + 1.14i)9-s + (1.14 + 0.0628i)10-s + (0.0636 + 0.110i)11-s + (1.51 + 0.166i)12-s + 1.46i·13-s + 1.74i·15-s + (−0.299 − 0.954i)16-s + (−0.949 − 1.64i)17-s + (−0.0721 + 1.31i)18-s + (0.330 − 0.572i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.575i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.818 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.818 - 0.575i$
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: \((2,\ 392,\ (\ :1),\ 0.818 - 0.575i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.95429 + 1.25074i\)
\(L(\frac12)\) \(\approx\) \(3.95429 + 1.25074i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.78 + 0.903i)T \)
7 \( 1 \)
good3 \( 1 + (-2.28 - 3.95i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-4.96 - 2.86i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-0.700 - 1.21i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 19.0iT - 169T^{2} \)
17 \( 1 + (16.1 + 27.9i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-6.28 + 10.8i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (13.7 + 7.94i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 3.29iT - 841T^{2} \)
31 \( 1 + (19.6 - 11.3i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-46.8 - 27.0i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 7.59T + 1.68e3T^{2} \)
43 \( 1 + 20.8T + 1.84e3T^{2} \)
47 \( 1 + (-18.7 - 10.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-0.308 + 0.178i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-13.4 - 23.2i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-74.6 - 43.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (57.2 + 99.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 104. iT - 5.04e3T^{2} \)
73 \( 1 + (12.1 + 21.1i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (101. + 58.5i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 79.2T + 6.88e3T^{2} \)
89 \( 1 + (-1.33 + 2.30i)T + (-3.96e3 - 6.85e3i)T^{2} \)
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

−11.13580509012977976889677443467, −10.21012339426933564775531664307, −9.514178731094551794155875992696, −9.030494930616438624459154501847, −7.12775757016009091870317459793, −6.23016175677147158897345810967, −4.90257223091058002982316519602, −4.26162409299363743055061582858, −2.96159575565441940149843456744, −2.16521030568715761176671445527, 1.56046746070257300672003763719, 2.54353436039945922888048098152, 3.89139929559395633854928497578, 5.58071378006553642847792240305, 6.05456624020385791691535861877, 7.21186783835102751903493145538, 8.119465907301549194243646362373, 8.660512406203560388868853224607, 10.04866024319624287023114067842, 11.29607542540865298868495954306

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