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Properties

Label 2-379050-1.1-c1-0-19
Degree $2$
Conductor $379050$
Sign $1$
Analytic cond. $3026.72$
Root an. cond. $55.0157$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 11-s − 12-s − 2·13-s + 14-s + 16-s + 5·17-s − 18-s + 21-s + 22-s + 24-s + 2·26-s − 27-s − 28-s + 4·29-s + 2·31-s − 32-s + 33-s − 5·34-s + 36-s + 4·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.21·17-s − 0.235·18-s + 0.218·21-s + 0.213·22-s + 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.188·28-s + 0.742·29-s + 0.359·31-s − 0.176·32-s + 0.174·33-s − 0.857·34-s + 1/6·36-s + 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(379050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(3026.72\)
Root analytic conductor: \(55.0157\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 379050,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7509364623\)
\(L(\frac12)\) \(\approx\) \(0.7509364623\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
19 \( 1 \)
good11 \( 1 + T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−12.36663923549684, −11.91705585502131, −11.67718425768972, −10.99433351418619, −10.64863190861622, −10.16445185645933, −9.764382098189862, −9.518031310132484, −8.910637677948491, −8.278703981931772, −7.882851789253676, −7.572306498140257, −6.907786002902359, −6.551145126246314, −6.060908512678506, −5.603063962557546, −4.955209188907010, −4.713240129726622, −3.873658416250917, −3.312806373716484, −2.842599227116785, −2.261569229182581, −1.509077520803341, −1.012451305813521, −0.2976111722629184, 0.2976111722629184, 1.012451305813521, 1.509077520803341, 2.261569229182581, 2.842599227116785, 3.312806373716484, 3.873658416250917, 4.713240129726622, 4.955209188907010, 5.603063962557546, 6.060908512678506, 6.551145126246314, 6.907786002902359, 7.572306498140257, 7.882851789253676, 8.278703981931772, 8.910637677948491, 9.518031310132484, 9.764382098189862, 10.16445185645933, 10.64863190861622, 10.99433351418619, 11.67718425768972, 11.91705585502131, 12.36663923549684

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