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Properties

Label 2-379050-1.1-c1-0-135
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 − 5·11-s − 12-s + 13-s − 14-s + 16-s + 5·17-s + 18-s + 21-s − 5·22-s − 24-s + 26-s − 27-s − 28-s − 6·29-s − 6·31-s + 32-s + 5·33-s + 5·34-s + 36-s − 8·37-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 − 1.50·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.21·17-s + 0.235·18-s + 0.218·21-s − 1.06·22-s − 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s − 1.07·31-s + 0.176·32-s + 0.870·33-s + 0.857·34-s + 1/6·36-s − 1.31·37-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: \(1\)
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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 7 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.73988966471563, −12.37817195463456, −11.91659129902621, −11.26442325708008, −11.00352739430581, −10.59101683441580, −10.00711846328325, −9.825473263271116, −9.123880635657001, −8.550645809946687, −7.979050603802874, −7.520724034175689, −7.215712663299469, −6.670233872403878, −6.030699138999172, −5.586657227778405, −5.325100064602057, −4.979818902494536, −4.155388290817432, −3.697949087104618, −3.303685656296746, −2.633546592286630, −2.127377705618242, −1.472564187107583, −0.6964192327145182, 0, 0.6964192327145182, 1.472564187107583, 2.127377705618242, 2.633546592286630, 3.303685656296746, 3.697949087104618, 4.155388290817432, 4.979818902494536, 5.325100064602057, 5.586657227778405, 6.030699138999172, 6.670233872403878, 7.215712663299469, 7.520724034175689, 7.979050603802874, 8.550645809946687, 9.123880635657001, 9.825473263271116, 10.00711846328325, 10.59101683441580, 11.00352739430581, 11.26442325708008, 11.91659129902621, 12.37817195463456, 12.73988966471563

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