Properties

Label 2-190575-1.1-c1-0-102
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 7-s + 13-s + 4·16-s + 17-s − 7·19-s + 8·23-s − 2·28-s − 6·29-s + 9·37-s + 41-s + 8·43-s + 49-s − 2·52-s − 53-s + 14·59-s − 5·61-s − 8·64-s + 5·67-s − 2·68-s − 9·71-s + 7·73-s + 14·76-s − 10·79-s + 8·83-s + 8·89-s + 91-s + ⋯
L(s)  = 1  − 4-s + 0.377·7-s + 0.277·13-s + 16-s + 0.242·17-s − 1.60·19-s + 1.66·23-s − 0.377·28-s − 1.11·29-s + 1.47·37-s + 0.156·41-s + 1.21·43-s + 1/7·49-s − 0.277·52-s − 0.137·53-s + 1.82·59-s − 0.640·61-s − 64-s + 0.610·67-s − 0.242·68-s − 1.06·71-s + 0.819·73-s + 1.60·76-s − 1.12·79-s + 0.878·83-s + 0.847·89-s + 0.104·91-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: \(1\)
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 \( 1 - T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−13.38109421977632, −12.86887662896832, −12.61670065638523, −12.03813741197178, −11.31559780914264, −10.92385283935262, −10.69150071689796, −9.887292532228544, −9.575352214064379, −8.938300875636275, −8.719027857302978, −8.193613859575464, −7.635746950892076, −7.223667349326651, −6.513135365361059, −5.975258758675756, −5.514633282791021, −4.938374567867676, −4.476247941789253, −3.980333447056012, −3.540838841993761, −2.722642447079656, −2.221458405155250, −1.317893848768237, −0.8283304673150226, 0, 0.8283304673150226, 1.317893848768237, 2.221458405155250, 2.722642447079656, 3.540838841993761, 3.980333447056012, 4.476247941789253, 4.938374567867676, 5.514633282791021, 5.975258758675756, 6.513135365361059, 7.223667349326651, 7.635746950892076, 8.193613859575464, 8.719027857302978, 8.938300875636275, 9.575352214064379, 9.887292532228544, 10.69150071689796, 10.92385283935262, 11.31559780914264, 12.03813741197178, 12.61670065638523, 12.86887662896832, 13.38109421977632

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