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Properties

Label 2-392-56.11-c2-0-21
Degree $2$
Conductor $392$
Sign $0.947 - 0.318i$
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 + (1.29 − 2.23i)3-s + (−1.99 + 3.46i)4-s − 5.17·6-s + 7.99·8-s + (1.15 + 2.00i)9-s + (−8.48 + 14.6i)11-s + (5.17 + 8.95i)12-s + (−8 − 13.8i)16-s + (−12.7 + 22.0i)17-s + (2.31 − 4.00i)18-s + (6.02 + 10.4i)19-s + 33.9·22-s + (10.3 − 17.9i)24-s + (−12.5 + 21.6i)25-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.430 − 0.746i)3-s + (−0.499 + 0.866i)4-s − 0.861·6-s + 0.999·8-s + (0.128 + 0.222i)9-s + (−0.771 + 1.33i)11-s + (0.430 + 0.746i)12-s + (−0.5 − 0.866i)16-s + (−0.747 + 1.29i)17-s + (0.128 − 0.222i)18-s + (0.316 + 0.548i)19-s + 1.54·22-s + (0.430 − 0.746i)24-s + (−0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.947 - 0.318i$
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: \((2,\ 392,\ (\ :1),\ 0.947 - 0.318i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.04498 + 0.170902i\)
\(L(\frac12)\) \(\approx\) \(1.04498 + 0.170902i\)
\(L(2)\) 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 \( 1 + (-1.29 + 2.23i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (8.48 - 14.6i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 169T^{2} \)
17 \( 1 + (12.7 - 22.0i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.02 - 10.4i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (264.5 - 458. i)T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 + (480.5 + 832. i)T^{2} \)
37 \( 1 + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 80.5T + 1.68e3T^{2} \)
43 \( 1 + 84.8T + 1.84e3T^{2} \)
47 \( 1 + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-58.9 + 102. i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-31 + 53.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + (62.2 - 107. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 75.7T + 6.88e3T^{2} \)
89 \( 1 + (-87.6 - 151. i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 186.T + 9.40e3T^{2} \)
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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

−11.03423877855728951268428439445, −10.23693428964759171459169836103, −9.442070333327624339409026825498, −8.239608424028170357308338550312, −7.73382042021032985570063858541, −6.78347578415336426643736344869, −5.05615861474197311342395761433, −3.87638609687681699126631135566, −2.39118321303827543702994653678, −1.60960572033065828612645286680, 0.52799127803862371742135392888, 2.83294581089423635248097748835, 4.26587316886841754349817585687, 5.26336379225767380223907298508, 6.33783461505365713818832162823, 7.39394476225324151962614601455, 8.449165062357427449071925977342, 9.080791563516029839968230344556, 9.905357654245880725016152202479, 10.73401538059195886158282049827

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