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Properties

Label 2-379050-1.1-c1-0-97
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 + 5·13-s + 14-s + 16-s + 17-s − 18-s + 21-s + 2·22-s − 5·23-s + 24-s − 5·26-s − 27-s − 28-s − 9·29-s − 3·31-s − 32-s + 2·33-s − 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 + 1.38·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.218·21-s + 0.426·22-s − 1.04·23-s + 0.204·24-s − 0.980·26-s − 0.192·27-s − 0.188·28-s − 1.67·29-s − 0.538·31-s − 0.176·32-s + 0.348·33-s − 0.171·34-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 \( 1 - 5 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−12.67569917407093, −12.19960621339259, −11.64727731579298, −11.32462642862492, −10.73683778659852, −10.52806216799061, −10.16172868172191, −9.408266138608449, −9.118644876629373, −8.808946827246300, −7.954758201873509, −7.724899473076252, −7.404420856915531, −6.584689087988735, −6.254888659444506, −5.868950206108127, −5.426661587342571, −4.838759852493704, −4.132475228062703, −3.526249508214859, −3.343474693262495, −2.388402170499342, −1.877293197384380, −1.362388520955160, −0.5926058654586778, 0, 0.5926058654586778, 1.362388520955160, 1.877293197384380, 2.388402170499342, 3.343474693262495, 3.526249508214859, 4.132475228062703, 4.838759852493704, 5.426661587342571, 5.868950206108127, 6.254888659444506, 6.584689087988735, 7.404420856915531, 7.724899473076252, 7.954758201873509, 8.808946827246300, 9.118644876629373, 9.408266138608449, 10.16172868172191, 10.52806216799061, 10.73683778659852, 11.32462642862492, 11.64727731579298, 12.19960621339259, 12.67569917407093

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