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Properties

Label 2-392-56.5-c2-0-75
Degree $2$
Conductor $392$
Sign $-0.0633 - 0.997i$
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 − 1.73i)2-s + (−2.64 − 4.58i)3-s + (−1.99 − 3.46i)4-s + (2.64 − 4.58i)5-s − 10.5·6-s − 7.99·8-s + (−9.5 + 16.4i)9-s + (−5.29 − 9.16i)10-s + (−10.5 + 18.3i)12-s − 5.29·13-s − 28·15-s + (−8 + 13.8i)16-s + (18.9 + 32.9i)18-s + (18.5 − 32.0i)19-s − 21.1·20-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.881 − 1.52i)3-s + (−0.499 − 0.866i)4-s + (0.529 − 0.916i)5-s − 1.76·6-s − 0.999·8-s + (−1.05 + 1.82i)9-s + (−0.529 − 0.916i)10-s + (−0.881 + 1.52i)12-s − 0.407·13-s − 1.86·15-s + (−0.5 + 0.866i)16-s + (1.05 + 1.82i)18-s + (0.974 − 1.68i)19-s − 1.05·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.0633 - 0.997i$
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: \((2,\ 392,\ (\ :1),\ -0.0633 - 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.809219 + 0.862200i\)
\(L(\frac12)\) \(\approx\) \(0.809219 + 0.862200i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
7 \( 1 \)
good3 \( 1 + (2.64 + 4.58i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-2.64 + 4.58i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (60.5 - 104. i)T^{2} \)
13 \( 1 + 5.29T + 169T^{2} \)
17 \( 1 + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-18.5 + 32.0i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-5 + 8.66i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 + (480.5 - 832. i)T^{2} \)
37 \( 1 + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 - 1.84e3T^{2} \)
47 \( 1 + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (44.9 + 77.9i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (60.8 - 105. i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 110T + 5.04e3T^{2} \)
73 \( 1 + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-65 + 112. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 164.T + 6.88e3T^{2} \)
89 \( 1 + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 9.40e3T^{2} \)
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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

−10.84554107033978529214998328939, −9.584053302229935320682366052470, −8.710167817302543824093811770944, −7.37134108216177719401179981236, −6.38709576327152776134008759259, −5.39135033415583032956120601449, −4.79271855611896192282078345193, −2.69840587790437401832712133676, −1.49700107879195959094106057320, −0.51122519803010860609308959567, 3.10187021243497380776927744576, 4.02971211421971657115476753725, 5.17937654924685074631536851737, 5.83617269227542443630907616801, 6.69311636398122918961397767335, 7.914275509570123204313663480211, 9.316005304376283199023950066991, 9.918716184787384710475778267183, 10.74168125894468922528805798832, 11.70607785522489109540946179412

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