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Properties

Label 2-392-56.51-c0-0-1
Degree $2$
Conductor $392$
Sign $0.947 + 0.318i$
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 + (0.707 + 1.22i)3-s + (−0.499 − 0.866i)4-s + 1.41·6-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.707 − 1.22i)12-s + (−0.5 + 0.866i)16-s + (−0.707 − 1.22i)17-s + (0.5 + 0.866i)18-s + (−0.707 + 1.22i)19-s + (−0.707 − 1.22i)24-s + (−0.5 − 0.866i)25-s + (0.499 + 0.866i)32-s − 1.41·34-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.707 + 1.22i)3-s + (−0.499 − 0.866i)4-s + 1.41·6-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.707 − 1.22i)12-s + (−0.5 + 0.866i)16-s + (−0.707 − 1.22i)17-s + (0.5 + 0.866i)18-s + (−0.707 + 1.22i)19-s + (−0.707 − 1.22i)24-s + (−0.5 − 0.866i)25-s + (0.499 + 0.866i)32-s − 1.41·34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.947 + 0.318i$
Analytic conductor: \(0.195633\)
Root analytic conductor: \(0.442304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :0),\ 0.947 + 0.318i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.166623517\)
\(L(\frac12)\) \(\approx\) \(1.166623517\)
\(L(1)\) 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 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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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

−11.39093439597806997439417601180, −10.41476363770107547413858918359, −9.873265699529936988052333387250, −9.055542959058167545682038208745, −8.217643887931389091092843722523, −6.50684546958193927497430349016, −5.19064242793329841487128014132, −4.31058511544820050170153889786, −3.46186598769586487547194133605, −2.28883595509610354560748598200, 2.13228944372364409470001681880, 3.48796433458912739743561166380, 4.81185762247012590249160734986, 6.20815541429223100426914943401, 6.85578042785168392983892937457, 7.75105913992276727584293806689, 8.507146198468054036463102140639, 9.226273715847906902359250565934, 10.82177021404789126881442254844, 11.99749138160849824781851778020

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