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Properties

Label 2-392-56.3-c1-0-21
Degree $2$
Conductor $392$
Sign $0.982 + 0.188i$
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  + (0.707 − 1.22i)2-s + (2.26 + 1.30i)3-s + (−0.999 − 1.73i)4-s + (1.60 + 2.77i)5-s + (3.20 − 1.84i)6-s − 2.82·8-s + (1.91 + 3.31i)9-s + 4.52·10-s + (1 − 1.73i)11-s − 5.22i·12-s − 5.07·13-s + 8.36i·15-s + (−2.00 + 3.46i)16-s + (−0.274 − 0.158i)17-s + 5.41·18-s + (4.13 − 2.38i)19-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)2-s + (1.30 + 0.754i)3-s + (−0.499 − 0.866i)4-s + (0.715 + 1.23i)5-s + (1.30 − 0.754i)6-s − 0.999·8-s + (0.638 + 1.10i)9-s + 1.43·10-s + (0.301 − 0.522i)11-s − 1.50i·12-s − 1.40·13-s + 2.15i·15-s + (−0.500 + 0.866i)16-s + (−0.0665 − 0.0384i)17-s + 1.27·18-s + (0.949 − 0.548i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.982 + 0.188i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ 0.982 + 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.57789 - 0.245184i\)
\(L(\frac12)\) \(\approx\) \(2.57789 - 0.245184i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
7 \( 1 \)
good3 \( 1 + (-2.26 - 1.30i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.60 - 2.77i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.07T + 13T^{2} \)
17 \( 1 + (0.274 + 0.158i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.13 + 2.38i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.24 - 0.717i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.37iT - 29T^{2} \)
31 \( 1 + (1.32 - 2.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.12 + 1.22i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.46iT - 41T^{2} \)
43 \( 1 + 8.82T + 43T^{2} \)
47 \( 1 + (-2.65 - 4.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 + 1.73i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.26 - 1.30i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.98 + 3.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-1.21 - 0.699i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.2 + 5.91i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.47iT - 83T^{2} \)
89 \( 1 + (8.55 - 4.93i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.82iT - 97T^{2} \)
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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

−11.14345304146768425141312721728, −10.11782974236006095126177230212, −9.762243428097121794785173080226, −8.992583829181240535130573484382, −7.65635210460627887068612488517, −6.40017531365964438297622233396, −5.14241972464519669882469495661, −3.89240958156620867380961484308, −2.90908270449910520266204144177, −2.30341511599795589282870391887, 1.76009892583259562989946231930, 3.14035176331194501768954292421, 4.61163912412591879440921713705, 5.46192513733399870189633544172, 6.83825530640370413275951557816, 7.61292530731853593554422228553, 8.426790571914545173429663147274, 9.239908442384282836371801840736, 9.785991101136904684804457599387, 12.02302396257065318408789039715

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