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Properties

Label 2-399-399.293-c0-0-0
Degree $2$
Conductor $399$
Sign $-0.0977 - 0.995i$
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  + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s − 0.999·12-s + (−0.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s + 19-s + (−0.499 − 0.866i)21-s + (0.5 + 0.866i)25-s + 0.999·27-s − 0.999·28-s + 31-s + (0.499 − 0.866i)36-s − 1.73i·37-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s − 0.999·12-s + (−0.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s + 19-s + (−0.499 − 0.866i)21-s + (0.5 + 0.866i)25-s + 0.999·27-s − 0.999·28-s + 31-s + (0.499 − 0.866i)36-s − 1.73i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(399\)    =    \(3 \cdot 7 \cdot 19\)
Sign: $-0.0977 - 0.995i$
Analytic conductor: \(0.199126\)
Root analytic conductor: \(0.446236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{399} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 399,\ (\ :0),\ -0.0977 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7371684953\)
\(L(\frac12)\) \(\approx\) \(0.7371684953\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 - T \)
good2 \( 1 + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
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 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + 1.73iT - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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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

−11.72465722849884184555307424374, −10.96019233253088897473532960778, −9.904053640622015743462921526270, −9.127534694388174600124712901249, −8.126019259460014878812871146585, −7.00488337267592848707182301154, −5.92022987538784082001969568899, −5.01560407538597719289788033501, −3.57272492612994412258823854923, −2.74540623192990583307490111422, 1.19629506160355462175283295401, 2.69094705467349542082355555945, 4.57028874424349059741946218838, 5.66331650678136141965320339859, 6.80595339897344670954338681743, 7.01202312709295540649604253045, 8.342560389871093536005179049531, 9.804071045336939630591414474186, 10.30880386432574066113567022219, 11.44220988600568209298668792885

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