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Properties

Label 2-399-399.254-c0-0-1
Degree $2$
Conductor $399$
Sign $0.365 + 0.930i$
Analytic cond. $0.199126$
Root an. cond. $0.446236$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + (0.866 + 0.5i)3-s − i·5-s + (0.5 − 0.866i)6-s − 7-s − i·8-s + (0.499 + 0.866i)9-s − 10-s + (−0.866 + 0.5i)11-s + (−0.5 + 0.866i)13-s + i·14-s + (0.5 − 0.866i)15-s − 16-s + (0.866 − 0.499i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  − i·2-s + (0.866 + 0.5i)3-s − i·5-s + (0.5 − 0.866i)6-s − 7-s − i·8-s + (0.499 + 0.866i)9-s − 10-s + (−0.866 + 0.5i)11-s + (−0.5 + 0.866i)13-s + i·14-s + (0.5 − 0.866i)15-s − 16-s + (0.866 − 0.499i)18-s + (0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(399\)    =    \(3 \cdot 7 \cdot 19\)
Sign: $0.365 + 0.930i$
Analytic conductor: \(0.199126\)
Root analytic conductor: \(0.446236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{399} (254, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 399,\ (\ :0),\ 0.365 + 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.049874095\)
\(L(\frac12)\) \(\approx\) \(1.049874095\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + T \)
19 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + iT - T^{2} \)
5 \( 1 + iT - T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−11.29763602780640139976436606580, −10.08790354982161575658004136579, −9.781327446989435104435129298373, −8.947806579146355483296697496095, −7.81874832968559315357843426579, −6.74411465903741999607492064175, −5.11886877477367550661698661009, −4.06792400464074187589834166117, −3.02766687561412748796023803433, −1.88503499254724993405033377265, 2.68651928957632541355170030889, 3.14713419398956792974648404091, 5.21837391304498984254199381890, 6.45663439494426011404886238950, 6.91955474325329710292631796935, 7.79433553150086254575625066452, 8.546870659230709247822226977547, 9.785998164501011030584694629824, 10.58253890788612866929130764149, 11.70906996247432913072797917300

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