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Properties

Label 2-392-7.4-c1-0-6
Degree $2$
Conductor $392$
Sign $0.605 + 0.795i$
Analytic cond. $3.13013$
Root an. cond. $1.76921$
Motivic weight $1$
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)5-s + (1.5 + 2.59i)9-s + (2 − 3.46i)11-s + 2·13-s + (3 − 5.19i)17-s + (−4 − 6.92i)19-s + (0.500 − 0.866i)25-s + 6·29-s + (−4 + 6.92i)31-s + (1 + 1.73i)37-s + 2·41-s − 4·43-s + (3 − 5.19i)45-s + (4 + 6.92i)47-s + (−3 + 5.19i)53-s + ⋯
L(s)  = 1  + (−0.447 − 0.774i)5-s + (0.5 + 0.866i)9-s + (0.603 − 1.04i)11-s + 0.554·13-s + (0.727 − 1.26i)17-s + (−0.917 − 1.58i)19-s + (0.100 − 0.173i)25-s + 1.11·29-s + (−0.718 + 1.24i)31-s + (0.164 + 0.284i)37-s + 0.312·41-s − 0.609·43-s + (0.447 − 0.774i)45-s + (0.583 + 1.01i)47-s + (−0.412 + 0.713i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18462 - 0.587249i\)
\(L(\frac12)\) \(\approx\) \(1.18462 - 0.587249i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8 + 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−11.19971378739028603905201432941, −10.40469223640186487596893712487, −9.069022381431254229018538250769, −8.567313669073813564430591680918, −7.50902632612095066162361842807, −6.46224666431223096370456714252, −5.12967271343095858206091713244, −4.36449030072195860083032903856, −2.91873216452083230725512049416, −1.00222359229110157409557222164, 1.70575969712221466509363245223, 3.57096991569393319230331323260, 4.14973688010729382898161935693, 5.94211590686605898519012397209, 6.69145821398429487272474581386, 7.64423184874793286340285178526, 8.629841428112019665094923856396, 9.852428124813606514694109476450, 10.39261740599788675139683158107, 11.48027091577279013636710047891

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