Properties

Label 2-91035-1.1-c1-0-15
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 5-s − 7-s + 2·11-s − 5·13-s + 4·16-s + 2·19-s + 2·20-s − 23-s + 25-s + 2·28-s + 8·29-s − 31-s + 35-s + 3·37-s − 7·41-s − 4·44-s + 47-s + 49-s + 10·52-s + 8·53-s − 2·55-s + 7·61-s − 8·64-s + 5·65-s + 16·67-s − 10·71-s + ⋯
L(s)  = 1  − 4-s − 0.447·5-s − 0.377·7-s + 0.603·11-s − 1.38·13-s + 16-s + 0.458·19-s + 0.447·20-s − 0.208·23-s + 1/5·25-s + 0.377·28-s + 1.48·29-s − 0.179·31-s + 0.169·35-s + 0.493·37-s − 1.09·41-s − 0.603·44-s + 0.145·47-s + 1/7·49-s + 1.38·52-s + 1.09·53-s − 0.269·55-s + 0.896·61-s − 64-s + 0.620·65-s + 1.95·67-s − 1.18·71-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 91035,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.402611150\)
\(L(\frac12)\) \(\approx\) \(1.402611150\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 13 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−13.94554207437811, −13.32460622629907, −12.91187043700990, −12.32469964963852, −11.89636581667159, −11.71971794022635, −10.72802283724803, −10.31095838501694, −9.733741350035130, −9.490403621016053, −8.877547717762594, −8.326794528678977, −7.923309337172661, −7.273721306051566, −6.782992210881994, −6.247247564860200, −5.406891446164450, −5.044151323841513, −4.500199780739346, −3.917682642143421, −3.420262791880752, −2.748539075937403, −2.037474596247330, −0.9754547923066802, −0.4766009314775503, 0.4766009314775503, 0.9754547923066802, 2.037474596247330, 2.748539075937403, 3.420262791880752, 3.917682642143421, 4.500199780739346, 5.044151323841513, 5.406891446164450, 6.247247564860200, 6.782992210881994, 7.273721306051566, 7.923309337172661, 8.326794528678977, 8.877547717762594, 9.490403621016053, 9.733741350035130, 10.31095838501694, 10.72802283724803, 11.71971794022635, 11.89636581667159, 12.32469964963852, 12.91187043700990, 13.32460622629907, 13.94554207437811

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