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Properties

Label 2-392-7.4-c1-0-3
Degree $2$
Conductor $392$
Sign $0.701 - 0.712i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.701 - 0.712i$
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: \((2,\ 392,\ (\ :1/2),\ 0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38639 + 0.581016i\)
\(L(\frac12)\) \(\approx\) \(1.38639 + 0.581016i\)
\(L(\frac{3}{2})\) 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

−11.32043421468784366133231299063, −10.36785676688309124009573448310, −9.929124609426543570709263219077, −8.539230659751575763907655294321, −7.76044616594718079628312705115, −6.54780586936613037623635497675, −5.86187912253282319375876365814, −4.46263616498577605798171719832, −3.19527635429439106026589972181, −1.80138803823286742340519672064, 1.15134921379481204045332394617, 2.82041761037243572874998032881, 4.52405899190915212718345758041, 5.06525128929975497774589015480, 6.71110751894114905772863100706, 7.14589684481559570253666464729, 8.725288688985365401154026504464, 9.439663427729552651315334271770, 9.917274417527311841030956296240, 11.40344511286972910858701886830

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