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Properties

Label 2-392-392.109-c1-0-25
Degree $2$
Conductor $392$
Sign $0.903 - 0.427i$
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  + (0.429 + 1.34i)2-s + (−0.197 − 1.31i)3-s + (−1.63 + 1.15i)4-s + (−0.400 − 0.157i)5-s + (1.68 − 0.831i)6-s + (2.29 − 1.31i)7-s + (−2.26 − 1.69i)8-s + (1.18 − 0.364i)9-s + (0.0396 − 0.606i)10-s + (−1.11 + 3.62i)11-s + (1.84 + 1.91i)12-s + (3.49 − 0.797i)13-s + (2.76 + 2.52i)14-s + (−0.127 + 0.556i)15-s + (1.31 − 3.77i)16-s + (4.88 − 3.32i)17-s + ⋯
L(s)  = 1  + (0.303 + 0.952i)2-s + (−0.114 − 0.758i)3-s + (−0.815 + 0.579i)4-s + (−0.178 − 0.0702i)5-s + (0.687 − 0.339i)6-s + (0.866 − 0.498i)7-s + (−0.799 − 0.600i)8-s + (0.393 − 0.121i)9-s + (0.0125 − 0.191i)10-s + (−0.337 + 1.09i)11-s + (0.532 + 0.552i)12-s + (0.968 − 0.221i)13-s + (0.738 + 0.674i)14-s + (−0.0328 + 0.143i)15-s + (0.329 − 0.944i)16-s + (1.18 − 0.807i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.903 - 0.427i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ 0.903 - 0.427i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53295 + 0.344622i\)
\(L(\frac12)\) \(\approx\) \(1.53295 + 0.344622i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.429 - 1.34i)T \)
7 \( 1 + (-2.29 + 1.31i)T \)
good3 \( 1 + (0.197 + 1.31i)T + (-2.86 + 0.884i)T^{2} \)
5 \( 1 + (0.400 + 0.157i)T + (3.66 + 3.40i)T^{2} \)
11 \( 1 + (1.11 - 3.62i)T + (-9.08 - 6.19i)T^{2} \)
13 \( 1 + (-3.49 + 0.797i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (-4.88 + 3.32i)T + (6.21 - 15.8i)T^{2} \)
19 \( 1 + (-1.60 + 0.928i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.85 - 4.67i)T + (8.40 + 21.4i)T^{2} \)
29 \( 1 + (1.43 - 2.97i)T + (-18.0 - 22.6i)T^{2} \)
31 \( 1 + (3.59 - 6.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.975 - 0.0731i)T + (36.5 - 5.51i)T^{2} \)
41 \( 1 + (1.32 + 1.66i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (5.52 + 4.40i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (1.01 - 0.938i)T + (3.51 - 46.8i)T^{2} \)
53 \( 1 + (-2.59 - 0.194i)T + (52.4 + 7.89i)T^{2} \)
59 \( 1 + (9.74 - 3.82i)T + (43.2 - 40.1i)T^{2} \)
61 \( 1 + (-9.85 + 0.738i)T + (60.3 - 9.09i)T^{2} \)
67 \( 1 + (11.4 + 6.62i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (13.6 - 6.55i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-5.75 - 5.34i)T + (5.45 + 72.7i)T^{2} \)
79 \( 1 + (4.67 + 8.10i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-16.5 - 3.77i)T + (74.7 + 36.0i)T^{2} \)
89 \( 1 + (6.38 - 1.96i)T + (73.5 - 50.1i)T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−11.70734403227620499094925213373, −10.40250547189960592549903599209, −9.333733596550388196728298577729, −8.200525678354588003082798308441, −7.31704381267557256207932170119, −7.06159941393934122644554043889, −5.56324692334910665059934035403, −4.74351600305770884278255609509, −3.48571736867769167614929248970, −1.29536935988864985723422583880, 1.47564103717050325398154246711, 3.20071247820525546143847124662, 4.10700361843658968180217817748, 5.22115925783676479431347968393, 5.94953861634616897133793869138, 7.87154860304033444036211923203, 8.702893015787190142353680370915, 9.616364927030770955080719148984, 10.65094141645731484384869205831, 11.10712587833417557665102871844

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