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Properties

Label 2-392-7.2-c1-0-9
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)3-s + (2 − 3.46i)5-s + (−0.499 + 0.866i)9-s − 7.99·15-s + (1 + 1.73i)17-s + (1 − 1.73i)19-s + (−4 + 6.92i)23-s + (−5.49 − 9.52i)25-s − 4.00·27-s + 2·29-s + (−2 − 3.46i)31-s + (3 − 5.19i)37-s − 2·41-s + 8·43-s + (1.99 + 3.46i)45-s + ⋯
L(s)  = 1  + (−0.577 − 0.999i)3-s + (0.894 − 1.54i)5-s + (−0.166 + 0.288i)9-s − 2.06·15-s + (0.242 + 0.420i)17-s + (0.229 − 0.397i)19-s + (−0.834 + 1.44i)23-s + (−1.09 − 1.90i)25-s − 0.769·27-s + 0.371·29-s + (−0.359 − 0.622i)31-s + (0.493 − 0.854i)37-s − 0.312·41-s + 1.21·43-s + (0.298 + 0.516i)45-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: $\chi_{392} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.551298 - 1.11210i\)
\(L(\frac12)\) \(\approx\) \(0.551298 - 1.11210i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2 + 3.46i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3 + 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (-2 + 3.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5 - 8.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−11.23285036828823323555144324239, −9.859162545584294594708466487086, −9.213776631061235487785156081965, −8.166102399136215303224807121059, −7.20560587115370710792019576716, −5.89617905358734567542833060655, −5.53675492852847305268946385047, −4.16126071596540159444221953544, −1.99503942564064382736952625075, −0.925685431336881532252264267021, 2.33708624739966318080004580333, 3.55683820040095531474443962281, 4.86117851223845245764222867140, 5.93037312169722568550874604180, 6.64816660922221670143116448464, 7.80608525003795736153474151825, 9.302809628078388426394940327673, 10.12280521541178391137915075361, 10.53419510080439711639113670991, 11.26038034934072338392547793775

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