Properties

Label 2-9075-1.1-c1-0-32
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  − 2.52·2-s + 3-s + 4.37·4-s − 2.52·6-s − 0.792·7-s − 5.98·8-s + 9-s + 4.37·12-s + 0.147·13-s + 2·14-s + 6.37·16-s − 6.63·17-s − 2.52·18-s + 4.40·19-s − 0.792·21-s − 8·23-s − 5.98·24-s − 0.372·26-s + 27-s − 3.46·28-s − 10.0·29-s + 2.37·31-s − 4.10·32-s + 16.7·34-s + 4.37·36-s − 5·37-s − 11.1·38-s + ⋯
L(s)  = 1  − 1.78·2-s + 0.577·3-s + 2.18·4-s − 1.03·6-s − 0.299·7-s − 2.11·8-s + 0.333·9-s + 1.26·12-s + 0.0409·13-s + 0.534·14-s + 1.59·16-s − 1.60·17-s − 0.594·18-s + 1.01·19-s − 0.172·21-s − 1.66·23-s − 1.22·24-s − 0.0730·26-s + 0.192·27-s − 0.654·28-s − 1.87·29-s + 0.426·31-s − 0.726·32-s + 2.87·34-s + 0.728·36-s − 0.821·37-s − 1.80·38-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

\(L(1)\) \(\approx\) \(0.6065919107\)
\(L(\frac12)\) \(\approx\) \(0.6065919107\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.52T + 2T^{2} \)
7 \( 1 + 0.792T + 7T^{2} \)
13 \( 1 - 0.147T + 13T^{2} \)
17 \( 1 + 6.63T + 17T^{2} \)
19 \( 1 - 4.40T + 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 + 10.0T + 29T^{2} \)
31 \( 1 - 2.37T + 31T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 + 6.92T + 41T^{2} \)
43 \( 1 + 9.94T + 43T^{2} \)
47 \( 1 - 8.74T + 47T^{2} \)
53 \( 1 - 1.25T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 5.98T + 61T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 - 5.19T + 73T^{2} \)
79 \( 1 - 6.78T + 79T^{2} \)
83 \( 1 + 8.51T + 83T^{2} \)
89 \( 1 - 5.48T + 89T^{2} \)
97 \( 1 - 9.37T + 97T^{2} \)
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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

−7.978427877034774668733723693244, −7.21986575544395339548918952224, −6.75133093364804486149728661259, −6.05579789125815759582819852960, −5.06550812826629806382288687223, −3.92743727211467745390012188302, −3.21438635880729828609354708176, −2.10844432305972024748007763575, −1.82689485134780157405556520597, −0.45827720590177441134170960819, 0.45827720590177441134170960819, 1.82689485134780157405556520597, 2.10844432305972024748007763575, 3.21438635880729828609354708176, 3.92743727211467745390012188302, 5.06550812826629806382288687223, 6.05579789125815759582819852960, 6.75133093364804486149728661259, 7.21986575544395339548918952224, 7.978427877034774668733723693244

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