Properties

Label 2-91035-1.1-c1-0-1
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 − 3·11-s + 5·13-s + 4·16-s + 2·19-s + 2·20-s − 6·23-s + 25-s + 2·28-s + 3·29-s + 4·31-s + 35-s − 2·37-s − 12·41-s − 10·43-s + 6·44-s − 9·47-s + 49-s − 10·52-s − 12·53-s + 3·55-s − 8·61-s − 8·64-s − 5·65-s − 4·67-s + ⋯
L(s)  = 1  − 4-s − 0.447·5-s − 0.377·7-s − 0.904·11-s + 1.38·13-s + 16-s + 0.458·19-s + 0.447·20-s − 1.25·23-s + 1/5·25-s + 0.377·28-s + 0.557·29-s + 0.718·31-s + 0.169·35-s − 0.328·37-s − 1.87·41-s − 1.52·43-s + 0.904·44-s − 1.31·47-s + 1/7·49-s − 1.38·52-s − 1.64·53-s + 0.404·55-s − 1.02·61-s − 64-s − 0.620·65-s − 0.488·67-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: \(0\)
Selberg data: \((2,\ 91035,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3087679584\)
\(L(\frac12)\) \(\approx\) \(0.3087679584\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−13.83552221093370, −13.29567572350189, −13.09125167707148, −12.42299370141560, −11.88931471905245, −11.51919190611806, −10.79025492235565, −10.22074742944370, −10.00922224709945, −9.400519699176070, −8.651766244710183, −8.385971028347069, −7.988272233924351, −7.453483263012786, −6.529624476238714, −6.264990924396531, −5.567282909200010, −4.926595898138212, −4.606761102089120, −3.803826167182396, −3.333631138912222, −2.988540137185805, −1.809293269241897, −1.225271155964847, −0.1917777132980869, 0.1917777132980869, 1.225271155964847, 1.809293269241897, 2.988540137185805, 3.333631138912222, 3.803826167182396, 4.606761102089120, 4.926595898138212, 5.567282909200010, 6.264990924396531, 6.529624476238714, 7.453483263012786, 7.988272233924351, 8.385971028347069, 8.651766244710183, 9.400519699176070, 10.00922224709945, 10.22074742944370, 10.79025492235565, 11.51919190611806, 11.88931471905245, 12.42299370141560, 13.09125167707148, 13.29567572350189, 13.83552221093370

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