Properties

Label 2-9075-1.1-c1-0-119
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  − 2.73·2-s + 3-s + 5.48·4-s − 2.73·6-s + 4.20·7-s − 9.52·8-s + 9-s + 5.48·12-s − 0.0784·13-s − 11.4·14-s + 15.0·16-s − 0.0228·17-s − 2.73·18-s − 1.26·19-s + 4.20·21-s + 6.12·23-s − 9.52·24-s + 0.214·26-s + 27-s + 23.0·28-s − 5.85·29-s − 3.23·31-s − 22.2·32-s + 0.0625·34-s + 5.48·36-s + 8.67·37-s + 3.46·38-s + ⋯
L(s)  = 1  − 1.93·2-s + 0.577·3-s + 2.74·4-s − 1.11·6-s + 1.58·7-s − 3.36·8-s + 0.333·9-s + 1.58·12-s − 0.0217·13-s − 3.07·14-s + 3.77·16-s − 0.00554·17-s − 0.644·18-s − 0.290·19-s + 0.917·21-s + 1.27·23-s − 1.94·24-s + 0.0420·26-s + 0.192·27-s + 4.35·28-s − 1.08·29-s − 0.581·31-s − 3.92·32-s + 0.0107·34-s + 0.913·36-s + 1.42·37-s + 0.561·38-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9075,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.390154954\)
\(L(\frac12)\) \(\approx\) \(1.390154954\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.73T + 2T^{2} \)
7 \( 1 - 4.20T + 7T^{2} \)
13 \( 1 + 0.0784T + 13T^{2} \)
17 \( 1 + 0.0228T + 17T^{2} \)
19 \( 1 + 1.26T + 19T^{2} \)
23 \( 1 - 6.12T + 23T^{2} \)
29 \( 1 + 5.85T + 29T^{2} \)
31 \( 1 + 3.23T + 31T^{2} \)
37 \( 1 - 8.67T + 37T^{2} \)
41 \( 1 + 3.84T + 41T^{2} \)
43 \( 1 + 0.859T + 43T^{2} \)
47 \( 1 - 1.89T + 47T^{2} \)
53 \( 1 - 4.55T + 53T^{2} \)
59 \( 1 + 3.44T + 59T^{2} \)
61 \( 1 - 8.13T + 61T^{2} \)
67 \( 1 + 0.406T + 67T^{2} \)
71 \( 1 - 8.40T + 71T^{2} \)
73 \( 1 - 15.1T + 73T^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 + 14.4T + 83T^{2} \)
89 \( 1 + 8.19T + 89T^{2} \)
97 \( 1 - 1.64T + 97T^{2} \)
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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

−7.989480096010720550902968520628, −7.29174910740035528675359046541, −6.92599994020504301902429100413, −5.86919724283891200750242915591, −5.12525870593927028768742179808, −4.07447913769579871835433594692, −3.01044701392859733786611863515, −2.19098667361783406337440400324, −1.61422946986895113400257982068, −0.77482604133053368260672542246, 0.77482604133053368260672542246, 1.61422946986895113400257982068, 2.19098667361783406337440400324, 3.01044701392859733786611863515, 4.07447913769579871835433594692, 5.12525870593927028768742179808, 5.86919724283891200750242915591, 6.92599994020504301902429100413, 7.29174910740035528675359046541, 7.989480096010720550902968520628

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