Properties

Label 2-190575-1.1-c1-0-129
Degree $2$
Conductor $190575$
Sign $1$
Analytic cond. $1521.74$
Root an. cond. $39.0096$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 7-s − 4·13-s + 4·16-s − 3·17-s + 19-s − 3·23-s − 2·28-s − 9·29-s − 10·31-s + 4·37-s + 6·41-s − 43-s − 6·47-s + 49-s + 8·52-s − 9·53-s − 15·59-s + 13·61-s − 8·64-s − 2·67-s + 6·68-s + 6·71-s − 16·73-s − 2·76-s − 8·79-s + 3·83-s + ⋯
L(s)  = 1  − 4-s + 0.377·7-s − 1.10·13-s + 16-s − 0.727·17-s + 0.229·19-s − 0.625·23-s − 0.377·28-s − 1.67·29-s − 1.79·31-s + 0.657·37-s + 0.937·41-s − 0.152·43-s − 0.875·47-s + 1/7·49-s + 1.10·52-s − 1.23·53-s − 1.95·59-s + 1.66·61-s − 64-s − 0.244·67-s + 0.727·68-s + 0.712·71-s − 1.87·73-s − 0.229·76-s − 0.900·79-s + 0.329·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190575\)    =    \(3^{2} \cdot 5^{2} \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1521.74\)
Root analytic conductor: \(39.0096\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 190575,\ (\ :1/2),\ 1)\)

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)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good2 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−13.50158423539662, −13.00816966997239, −12.77217333959893, −12.34006268867289, −11.61923111432442, −11.25797594033849, −10.81091958556659, −10.17654868647736, −9.611234361696056, −9.435444524241495, −8.876665593227748, −8.441774291577831, −7.636719392573735, −7.592411520910822, −7.022643612645272, −6.075398820058455, −5.827264422376153, −5.181046603219010, −4.665194468421556, −4.384125203927281, −3.617167325018791, −3.276224517034324, −2.306139624853560, −1.902964549394980, −1.142114970044820, 0, 0, 1.142114970044820, 1.902964549394980, 2.306139624853560, 3.276224517034324, 3.617167325018791, 4.384125203927281, 4.665194468421556, 5.181046603219010, 5.827264422376153, 6.075398820058455, 7.022643612645272, 7.592411520910822, 7.636719392573735, 8.441774291577831, 8.876665593227748, 9.435444524241495, 9.611234361696056, 10.17654868647736, 10.81091958556659, 11.25797594033849, 11.61923111432442, 12.34006268867289, 12.77217333959893, 13.00816966997239, 13.50158423539662

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