Properties

Label 2-9075-1.1-c1-0-0
Degree $2$
Conductor $9075$
Sign $1$
Analytic cond. $72.4642$
Root an. cond. $8.51259$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.792·2-s + 3-s − 1.37·4-s − 0.792·6-s − 2.52·7-s + 2.67·8-s + 9-s − 1.37·12-s − 6.78·13-s + 2·14-s + 0.627·16-s − 6.63·17-s − 0.792·18-s − 7.72·19-s − 2.52·21-s − 8·23-s + 2.67·24-s + 5.37·26-s + 27-s + 3.46·28-s − 3.16·29-s − 3.37·31-s − 5.84·32-s + 5.25·34-s − 1.37·36-s − 5·37-s + 6.11·38-s + ⋯
L(s)  = 1  − 0.560·2-s + 0.577·3-s − 0.686·4-s − 0.323·6-s − 0.954·7-s + 0.944·8-s + 0.333·9-s − 0.396·12-s − 1.88·13-s + 0.534·14-s + 0.156·16-s − 1.60·17-s − 0.186·18-s − 1.77·19-s − 0.550·21-s − 1.66·23-s + 0.545·24-s + 1.05·26-s + 0.192·27-s + 0.654·28-s − 0.588·29-s − 0.605·31-s − 1.03·32-s + 0.901·34-s − 0.228·36-s − 0.821·37-s + 0.992·38-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: no
Arithmetic: yes
Character: $\chi_{9075} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9075,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.04357641748\)
\(L(\frac12)\) \(\approx\) \(0.04357641748\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.792T + 2T^{2} \)
7 \( 1 + 2.52T + 7T^{2} \)
13 \( 1 + 6.78T + 13T^{2} \)
17 \( 1 + 6.63T + 17T^{2} \)
19 \( 1 + 7.72T + 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 + 3.16T + 29T^{2} \)
31 \( 1 + 3.37T + 31T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 - 6.92T + 41T^{2} \)
43 \( 1 + 9.94T + 43T^{2} \)
47 \( 1 + 2.74T + 47T^{2} \)
53 \( 1 - 12.7T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 2.67T + 61T^{2} \)
67 \( 1 + 6.11T + 67T^{2} \)
71 \( 1 + 0.744T + 71T^{2} \)
73 \( 1 + 5.19T + 73T^{2} \)
79 \( 1 + 0.147T + 79T^{2} \)
83 \( 1 - 1.87T + 83T^{2} \)
89 \( 1 + 17.4T + 89T^{2} \)
97 \( 1 - 3.62T + 97T^{2} \)
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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

−7.80249653896162231380920184441, −7.16094849275382881948016584935, −6.60016766105726005045671360950, −5.71428375770381688300030498108, −4.70398829981743789645496452258, −4.25879365006697737128370813263, −3.53465375237912974058674869952, −2.32913189735522683948155299874, −1.98602178282388334004510061181, −0.098829654736545109562797906734, 0.098829654736545109562797906734, 1.98602178282388334004510061181, 2.32913189735522683948155299874, 3.53465375237912974058674869952, 4.25879365006697737128370813263, 4.70398829981743789645496452258, 5.71428375770381688300030498108, 6.60016766105726005045671360950, 7.16094849275382881948016584935, 7.80249653896162231380920184441

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