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Properties

Label 2-392-56.11-c2-0-18
Degree $2$
Conductor $392$
Sign $-0.907 - 0.420i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.907 - 0.420i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ -0.907 - 0.420i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.138501 + 0.628576i\)
\(L(\frac12)\) \(\approx\) \(0.138501 + 0.628576i\)
\(L(2)\) 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

−11.15531843991969165302742930058, −10.36512043498307128485508398254, −9.862165508246873753436759201030, −8.824402662793784044015717485582, −8.049713881139818775665983429856, −6.70128718430214389900714762901, −5.82216862331398274668932944893, −4.81522367254571877651332481371, −3.34295201545710756990606671114, −1.53160985749395844380932973073, 0.42900727732236826752626436687, 1.72802823498389380240309949772, 3.02627804151330172621115424146, 5.02042945139244288121986384595, 6.42963572401096678345719171665, 6.94867388379874137763578029332, 7.70552334842967456743663851013, 9.047838595179449168940560744814, 9.611711697510231171295982052448, 10.83548929393936753363702013743

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