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Properties

Label 2-399-399.398-c0-0-4
Degree $2$
Conductor $399$
Sign $1$
Analytic cond. $0.199126$
Root an. cond. $0.446236$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s − 3-s + 1.00·4-s + 1.41·5-s − 1.41·6-s − 7-s + 9-s + 2.00·10-s − 1.00·12-s − 1.41·14-s − 1.41·15-s − 0.999·16-s − 1.41·17-s + 1.41·18-s + 19-s + 1.41·20-s + 21-s + 1.00·25-s − 27-s − 1.00·28-s − 1.41·29-s − 2.00·30-s − 1.41·32-s − 2.00·34-s − 1.41·35-s + 1.00·36-s + 1.41·38-s + ⋯
L(s)  = 1  + 1.41·2-s − 3-s + 1.00·4-s + 1.41·5-s − 1.41·6-s − 7-s + 9-s + 2.00·10-s − 1.00·12-s − 1.41·14-s − 1.41·15-s − 0.999·16-s − 1.41·17-s + 1.41·18-s + 19-s + 1.41·20-s + 21-s + 1.00·25-s − 27-s − 1.00·28-s − 1.41·29-s − 2.00·30-s − 1.41·32-s − 2.00·34-s − 1.41·35-s + 1.00·36-s + 1.41·38-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(399\)    =    \(3 \cdot 7 \cdot 19\)
Sign: $1$
Analytic conductor: \(0.199126\)
Root analytic conductor: \(0.446236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{399} (398, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 399,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.324631479\)
\(L(\frac12)\) \(\approx\) \(1.324631479\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + T \)
19 \( 1 - T \)
good2 \( 1 - 1.41T + T^{2} \)
5 \( 1 - 1.41T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + 1.41T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 - 1.41T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 2T + T^{2} \)
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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

−11.70882218034599812755458983043, −10.83148625236504410596301219625, −9.775176422830961415080156144109, −9.182575828780399421253333943894, −7.04373082961640695260063651005, −6.34187280635941636950486778730, −5.69032604357031132874129767075, −4.91791576780185149061613378859, −3.65462569966984243568578921039, −2.20793167252447025357418879335, 2.20793167252447025357418879335, 3.65462569966984243568578921039, 4.91791576780185149061613378859, 5.69032604357031132874129767075, 6.34187280635941636950486778730, 7.04373082961640695260063651005, 9.182575828780399421253333943894, 9.775176422830961415080156144109, 10.83148625236504410596301219625, 11.70882218034599812755458983043

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