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Properties

Label 2-399-399.110-c0-0-0
Degree $2$
Conductor $399$
Sign $-0.585 - 0.810i$
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  + (−0.766 + 0.642i)3-s + (−0.173 + 0.984i)4-s + (−0.939 + 0.342i)7-s + (0.173 − 0.984i)9-s + (−0.5 − 0.866i)12-s + (−0.326 + 1.85i)13-s + (−0.939 − 0.342i)16-s − 19-s + (0.5 − 0.866i)21-s + (0.939 − 0.342i)25-s + (0.500 + 0.866i)27-s + (−0.173 − 0.984i)28-s + 1.53·31-s + (0.939 + 0.342i)36-s + (1.70 + 0.984i)37-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)3-s + (−0.173 + 0.984i)4-s + (−0.939 + 0.342i)7-s + (0.173 − 0.984i)9-s + (−0.5 − 0.866i)12-s + (−0.326 + 1.85i)13-s + (−0.939 − 0.342i)16-s − 19-s + (0.5 − 0.866i)21-s + (0.939 − 0.342i)25-s + (0.500 + 0.866i)27-s + (−0.173 − 0.984i)28-s + 1.53·31-s + (0.939 + 0.342i)36-s + (1.70 + 0.984i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(399\)    =    \(3 \cdot 7 \cdot 19\)
Sign: $-0.585 - 0.810i$
Analytic conductor: \(0.199126\)
Root analytic conductor: \(0.446236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{399} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 399,\ (\ :0),\ -0.585 - 0.810i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5098247257\)
\(L(\frac12)\) \(\approx\) \(0.5098247257\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.766 - 0.642i)T \)
7 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + T \)
good2 \( 1 + (0.173 - 0.984i)T^{2} \)
5 \( 1 + (-0.939 + 0.342i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
17 \( 1 + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.766 - 0.642i)T^{2} \)
31 \( 1 - 1.53T + T^{2} \)
37 \( 1 + (-1.70 - 0.984i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.939 - 0.342i)T^{2} \)
43 \( 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.939 - 0.342i)T^{2} \)
53 \( 1 + (-0.939 - 0.342i)T^{2} \)
59 \( 1 + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (0.439 + 1.20i)T + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (-0.439 + 0.524i)T + (-0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.173 - 0.984i)T^{2} \)
73 \( 1 + (-0.439 - 0.524i)T + (-0.173 + 0.984i)T^{2} \)
79 \( 1 + (0.439 - 1.20i)T + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
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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

−11.86659116262260369093287250517, −11.08132644281398497784701044627, −9.841972604950179721925688965407, −9.241813082284408764596729718613, −8.325782184886548084124497698111, −6.75857442434159908319389917089, −6.40642193289514643874027728686, −4.73715340245241738788994312315, −4.05368792781209756199380944555, −2.73316157442354504973908410888, 0.78169091650337380349031349174, 2.68684156072379915195060162685, 4.52557202555545190941733793829, 5.62912528322123446992134405007, 6.27230168895615760336101874052, 7.21763010389751902209129431214, 8.335241189732926326185990298982, 9.663981819894624684256573503254, 10.45387672179324208923031345690, 10.89519750699449590781682960237

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