Properties

Label 2-91035-1.1-c1-0-31
Degree $2$
Conductor $91035$
Sign $-1$
Analytic cond. $726.918$
Root an. cond. $26.9614$
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  − 2·2-s + 2·4-s + 5-s − 7-s − 2·10-s − 2·11-s − 13-s + 2·14-s − 4·16-s + 2·19-s + 2·20-s + 4·22-s − 23-s + 25-s + 2·26-s − 2·28-s + 2·29-s + 5·31-s + 8·32-s − 35-s + 37-s − 4·38-s − 3·41-s + 4·43-s − 4·44-s + 2·46-s − 47-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.447·5-s − 0.377·7-s − 0.632·10-s − 0.603·11-s − 0.277·13-s + 0.534·14-s − 16-s + 0.458·19-s + 0.447·20-s + 0.852·22-s − 0.208·23-s + 1/5·25-s + 0.392·26-s − 0.377·28-s + 0.371·29-s + 0.898·31-s + 1.41·32-s − 0.169·35-s + 0.164·37-s − 0.648·38-s − 0.468·41-s + 0.609·43-s − 0.603·44-s + 0.294·46-s − 0.145·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91035\)    =    \(3^{2} \cdot 5 \cdot 7 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(726.918\)
Root analytic conductor: \(26.9614\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{91035} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 91035,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 + T \)
17 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−14.03215775423705, −13.60717466279909, −13.17501016207639, −12.53294354963310, −12.09165751882829, −11.43384094654009, −10.96392043397724, −10.42419956004421, −10.03461163757393, −9.640556739260207, −9.201626242939187, −8.644344719195919, −8.110429077057422, −7.704513178227391, −7.178988672605334, −6.553942180771195, −6.171361222818170, −5.377323357367373, −4.861482781281828, −4.254309197531025, −3.376923818433611, −2.703455080869668, −2.214566731003183, −1.446249517399580, −0.7787785065546461, 0, 0.7787785065546461, 1.446249517399580, 2.214566731003183, 2.703455080869668, 3.376923818433611, 4.254309197531025, 4.861482781281828, 5.377323357367373, 6.171361222818170, 6.553942180771195, 7.178988672605334, 7.704513178227391, 8.110429077057422, 8.644344719195919, 9.201626242939187, 9.640556739260207, 10.03461163757393, 10.42419956004421, 10.96392043397724, 11.43384094654009, 12.09165751882829, 12.53294354963310, 13.17501016207639, 13.60717466279909, 14.03215775423705

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