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Properties

Label 2-379050-1.1-c1-0-171
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 − 11-s − 12-s + 5·13-s + 14-s + 16-s + 7·17-s − 18-s + 21-s + 22-s + 4·23-s + 24-s − 5·26-s − 27-s − 28-s + 6·29-s + 2·31-s − 32-s + 33-s − 7·34-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 − 0.301·11-s − 0.288·12-s + 1.38·13-s + 0.267·14-s + 1/4·16-s + 1.69·17-s − 0.235·18-s + 0.218·21-s + 0.213·22-s + 0.834·23-s + 0.204·24-s − 0.980·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.359·31-s − 0.176·32-s + 0.174·33-s − 1.20·34-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 \( 1 + T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−12.41602477901035, −12.30267353568084, −11.71391026628288, −11.33730677138103, −10.76158826148245, −10.47092010894057, −10.04808707282489, −9.623156557265853, −9.134146738842306, −8.512176533982034, −8.220220057049563, −7.775865244854828, −7.161318940185714, −6.690622247628390, −6.324220706867182, −5.806738512855146, −5.377380480700180, −4.869127721590046, −4.198321498092098, −3.544990255937324, −3.092460551690851, −2.725722098111144, −1.659277753265653, −1.292999970142553, −0.7925421002435334, 0, 0.7925421002435334, 1.292999970142553, 1.659277753265653, 2.725722098111144, 3.092460551690851, 3.544990255937324, 4.198321498092098, 4.869127721590046, 5.377380480700180, 5.806738512855146, 6.324220706867182, 6.690622247628390, 7.161318940185714, 7.775865244854828, 8.220220057049563, 8.512176533982034, 9.134146738842306, 9.623156557265853, 10.04808707282489, 10.47092010894057, 10.76158826148245, 11.33730677138103, 11.71391026628288, 12.30267353568084, 12.41602477901035

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