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Properties

Label 2-392-56.5-c2-0-47
Degree $2$
Conductor $392$
Sign $0.998 - 0.0574i$
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 + 0.343i)2-s + (1.94 + 3.37i)3-s + (3.76 − 1.35i)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 + (−6.09 + 16.6i)10-s + (−3.15 + 1.82i)11-s + (11.8 + 10.0i)12-s + 7.79·13-s + 34.5·15-s + (12.3 − 10.1i)16-s + (9.07 − 5.23i)17-s + (4.25 − 11.6i)18-s + (−5.39 + 9.34i)19-s + ⋯
L(s)  = 1  + (−0.985 + 0.171i)2-s + (0.649 + 1.12i)3-s + (0.941 − 0.338i)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 + (−0.609 + 1.66i)10-s + (−0.287 + 0.165i)11-s + (0.991 + 0.838i)12-s + 0.599·13-s + 2.30·15-s + (0.771 − 0.636i)16-s + (0.533 − 0.308i)17-s + (0.236 − 0.644i)18-s + (−0.283 + 0.491i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0574i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.998 - 0.0574i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ 0.998 - 0.0574i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.64188 + 0.0471743i\)
\(L(\frac12)\) \(\approx\) \(1.64188 + 0.0471743i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.97 - 0.343i)T \)
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

−10.55766686111859496099883850928, −9.895717244010420931130380418594, −9.273776425380237449614136885909, −8.633896298595123303501960446358, −7.904006309041957774633252603620, −6.22903382391427540849405300733, −5.29198486661924334025726780640, −4.19638976718967764386554664276, −2.54886108048562680956481732143, −1.05767719737762592395308626968, 1.41406153666952258874081324997, 2.50611165908837339662119768132, 3.22520339094509020706074480461, 5.92050837562465309462193350984, 6.68874053901794049570632635384, 7.36371524058567156798034187377, 8.213179693688305519135595166176, 9.210332623515765399286462892795, 10.24572368012940589983062528750, 10.82824165362349013139204715953

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