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Properties

Label 2-392-56.45-c0-0-0
Degree $2$
Conductor $392$
Sign $0.0633 - 0.997i$
Analytic cond. $0.195633$
Root an. cond. $0.442304$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 0.999·8-s + (0.5 + 0.866i)9-s + (−0.5 − 0.866i)16-s + (−0.499 + 0.866i)18-s + (−1 − 1.73i)23-s + (0.5 − 0.866i)25-s + (0.499 − 0.866i)32-s − 0.999·36-s + (0.999 − 1.73i)46-s + 0.999·50-s + 0.999·64-s − 2·71-s + (−0.5 − 0.866i)72-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 0.999·8-s + (0.5 + 0.866i)9-s + (−0.5 − 0.866i)16-s + (−0.499 + 0.866i)18-s + (−1 − 1.73i)23-s + (0.5 − 0.866i)25-s + (0.499 − 0.866i)32-s − 0.999·36-s + (0.999 − 1.73i)46-s + 0.999·50-s + 0.999·64-s − 2·71-s + (−0.5 − 0.866i)72-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.0633 - 0.997i$
Analytic conductor: \(0.195633\)
Root analytic conductor: \(0.442304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :0),\ 0.0633 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9953880288\)
\(L(\frac12)\) \(\approx\) \(0.9953880288\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + 2T + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−12.05940743033006298842061213071, −10.79601194264097012401751112321, −9.906056305895301847157636311335, −8.648279835939016432195387230525, −7.978251735235705086361206145857, −6.97675941038514901831804631625, −6.08312938350553440014713552754, −4.90508162442482408190020139213, −4.11635898886893795513022385329, −2.53188825974446287111789836024, 1.56666874135914760959184313138, 3.21681588884546462201049839773, 4.11605887472660055035609750152, 5.35240122797776474584816552621, 6.32883466400891726875525612439, 7.52525234998560883881389246793, 8.951273619472734592430076120368, 9.628062291175097166386831973508, 10.45457744067465992267840288492, 11.50816697241054945858737934193

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