Properties

Label 2-9075-1.1-c1-0-314
Degree $2$
Conductor $9075$
Sign $-1$
Analytic cond. $72.4642$
Root an. cond. $8.51259$
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-s + 3-s − 4-s + 6-s + 4·7-s − 3·8-s + 9-s − 12-s − 2·13-s + 4·14-s − 16-s − 2·17-s + 18-s + 4·21-s − 8·23-s − 3·24-s − 2·26-s + 27-s − 4·28-s + 6·29-s − 8·31-s + 5·32-s − 2·34-s − 36-s − 6·37-s − 2·39-s + 2·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s − 0.288·12-s − 0.554·13-s + 1.06·14-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.872·21-s − 1.66·23-s − 0.612·24-s − 0.392·26-s + 0.192·27-s − 0.755·28-s + 1.11·29-s − 1.43·31-s + 0.883·32-s − 0.342·34-s − 1/6·36-s − 0.986·37-s − 0.320·39-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9075\)    =    \(3 \cdot 5^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(72.4642\)
Root analytic conductor: \(8.51259\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9075,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
11 \( 1 \)
good2 \( 1 - T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.56703394334927679521241514429, −6.64989431233589188309119242243, −5.81886729913280632057895570273, −5.08451211650611410303030375999, −4.58601163808162548244809882989, −4.01898082351391503616891777906, −3.19345176929607987672594852845, −2.23718821890583247191367747104, −1.52741118326423034153490846090, 0, 1.52741118326423034153490846090, 2.23718821890583247191367747104, 3.19345176929607987672594852845, 4.01898082351391503616891777906, 4.58601163808162548244809882989, 5.08451211650611410303030375999, 5.81886729913280632057895570273, 6.64989431233589188309119242243, 7.56703394334927679521241514429

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