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Properties

Label 2-392-8.3-c2-0-44
Degree $2$
Conductor $392$
Sign $-0.938 - 0.345i$
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  + (−0.789 − 1.83i)2-s − 5.33·3-s + (−2.75 + 2.90i)4-s − 2.15i·5-s + (4.21 + 9.79i)6-s + (7.50 + 2.76i)8-s + 19.4·9-s + (−3.96 + 1.70i)10-s + 5.25·11-s + (14.6 − 15.4i)12-s − 21.4i·13-s + 11.5i·15-s + (−0.841 − 15.9i)16-s − 0.926·17-s + (−15.3 − 35.7i)18-s + 5.93·19-s + ⋯
L(s)  = 1  + (−0.394 − 0.918i)2-s − 1.77·3-s + (−0.688 + 0.725i)4-s − 0.431i·5-s + (0.701 + 1.63i)6-s + (0.938 + 0.345i)8-s + 2.15·9-s + (−0.396 + 0.170i)10-s + 0.478·11-s + (1.22 − 1.28i)12-s − 1.64i·13-s + 0.766i·15-s + (−0.0525 − 0.998i)16-s − 0.0545·17-s + (−0.852 − 1.98i)18-s + 0.312·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.938 - 0.345i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ -0.938 - 0.345i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0623630 + 0.349420i\)
\(L(\frac12)\) \(\approx\) \(0.0623630 + 0.349420i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.789 + 1.83i)T \)
7 \( 1 \)
good3 \( 1 + 5.33T + 9T^{2} \)
5 \( 1 + 2.15iT - 25T^{2} \)
11 \( 1 - 5.25T + 121T^{2} \)
13 \( 1 + 21.4iT - 169T^{2} \)
17 \( 1 + 0.926T + 289T^{2} \)
19 \( 1 - 5.93T + 361T^{2} \)
23 \( 1 - 8.68iT - 529T^{2} \)
29 \( 1 + 9.42iT - 841T^{2} \)
31 \( 1 - 34.5iT - 961T^{2} \)
37 \( 1 + 12.8iT - 1.36e3T^{2} \)
41 \( 1 + 43.1T + 1.68e3T^{2} \)
43 \( 1 + 41.7T + 1.84e3T^{2} \)
47 \( 1 + 45.9iT - 2.20e3T^{2} \)
53 \( 1 + 74.4iT - 2.80e3T^{2} \)
59 \( 1 + 53.6T + 3.48e3T^{2} \)
61 \( 1 - 27.8iT - 3.72e3T^{2} \)
67 \( 1 + 78.4T + 4.48e3T^{2} \)
71 \( 1 + 74.5iT - 5.04e3T^{2} \)
73 \( 1 + 33.6T + 5.32e3T^{2} \)
79 \( 1 + 30.2iT - 6.24e3T^{2} \)
83 \( 1 - 72.9T + 6.88e3T^{2} \)
89 \( 1 - 54.8T + 7.92e3T^{2} \)
97 \( 1 - 53.7T + 9.40e3T^{2} \)
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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

−10.53254539105366812903373450581, −10.20039237846792239137137818096, −8.987877464600847039117035168577, −7.84054518841963336430659152234, −6.70736217262570970476560909012, −5.41480429847835312217406245289, −4.82579885726072380319989161756, −3.41994523252877762068505357563, −1.35850045939903106057005727299, −0.27187675342815799947151495075, 1.36845183561840673145036291373, 4.22722876037187727295900957907, 5.01282981360329385985651420417, 6.20792104709115171176121722953, 6.62604952485922917708092035405, 7.44608888930826914701464475753, 8.946229769789006687295484043563, 9.826208160524757891712081824027, 10.72234518461182388846202937528, 11.44639012997622995357046333814

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