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Properties

Label 2-379050-1.1-c1-0-153
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 $1$

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 − 2·11-s − 12-s + 3·13-s + 14-s + 16-s + 2·17-s − 18-s + 21-s + 2·22-s + 4·23-s + 24-s − 3·26-s − 27-s − 28-s − 4·29-s + 4·31-s − 32-s + 2·33-s − 2·34-s + 36-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 − 0.603·11-s − 0.288·12-s + 0.832·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.218·21-s + 0.426·22-s + 0.834·23-s + 0.204·24-s − 0.588·26-s − 0.192·27-s − 0.188·28-s − 0.742·29-s + 0.718·31-s − 0.176·32-s + 0.348·33-s − 0.342·34-s + 1/6·36-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: \(1\)
Selberg data: \((2,\ 379050,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 5 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−12.55472474588293, −12.26153184413027, −11.63367762674192, −11.32888433581780, −10.81816154020521, −10.46643891807950, −10.07831685510877, −9.498506427391929, −9.219878111287786, −8.574919811445567, −8.117964737953852, −7.816854903298709, −7.114205985607503, −6.710297545841886, −6.411327416436958, −5.745413405675762, −5.262408830534898, −5.022651223480175, −4.091339730093710, −3.659324434554134, −3.138247358275482, −2.498306286649804, −1.925521309747483, −1.179135597728606, −0.7399202913353517, 0, 0.7399202913353517, 1.179135597728606, 1.925521309747483, 2.498306286649804, 3.138247358275482, 3.659324434554134, 4.091339730093710, 5.022651223480175, 5.262408830534898, 5.745413405675762, 6.411327416436958, 6.710297545841886, 7.114205985607503, 7.816854903298709, 8.117964737953852, 8.574919811445567, 9.219878111287786, 9.498506427391929, 10.07831685510877, 10.46643891807950, 10.81816154020521, 11.32888433581780, 11.63367762674192, 12.26153184413027, 12.55472474588293

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