Properties

Label 2-9075-1.1-c1-0-79
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  − 1.61·2-s − 3-s + 0.618·4-s + 1.61·6-s − 5.23·7-s + 2.23·8-s + 9-s − 0.618·12-s + 3.23·13-s + 8.47·14-s − 4.85·16-s + 2·17-s − 1.61·18-s + 5·19-s + 5.23·21-s + 4.61·23-s − 2.23·24-s − 5.23·26-s − 27-s − 3.23·28-s + 0.854·29-s + 7·31-s + 3.38·32-s − 3.23·34-s + 0.618·36-s + 1.47·37-s − 8.09·38-s + ⋯
L(s)  = 1  − 1.14·2-s − 0.577·3-s + 0.309·4-s + 0.660·6-s − 1.97·7-s + 0.790·8-s + 0.333·9-s − 0.178·12-s + 0.897·13-s + 2.26·14-s − 1.21·16-s + 0.485·17-s − 0.381·18-s + 1.14·19-s + 1.14·21-s + 0.962·23-s − 0.456·24-s − 1.02·26-s − 0.192·27-s − 0.611·28-s + 0.158·29-s + 1.25·31-s + 0.597·32-s − 0.554·34-s + 0.103·36-s + 0.242·37-s − 1.31·38-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

\(L(1)\) \(\approx\) \(0.7359812756\)
\(L(\frac12)\) \(\approx\) \(0.7359812756\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.61T + 2T^{2} \)
7 \( 1 + 5.23T + 7T^{2} \)
13 \( 1 - 3.23T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 - 4.61T + 23T^{2} \)
29 \( 1 - 0.854T + 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 - 1.47T + 37T^{2} \)
41 \( 1 - 11.4T + 41T^{2} \)
43 \( 1 - 9.09T + 43T^{2} \)
47 \( 1 + 6.09T + 47T^{2} \)
53 \( 1 + 5.38T + 53T^{2} \)
59 \( 1 - 6.70T + 59T^{2} \)
61 \( 1 - 7T + 61T^{2} \)
67 \( 1 + 11.6T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 4.52T + 73T^{2} \)
79 \( 1 - 3.09T + 79T^{2} \)
83 \( 1 - 7.70T + 83T^{2} \)
89 \( 1 - 4.14T + 89T^{2} \)
97 \( 1 - 2.52T + 97T^{2} \)
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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

−7.57092483065320991340095813349, −7.26118850252987331100961143908, −6.30763414854500950073895718061, −6.05564317611295830637091442587, −5.06407135204977400579289229628, −4.14767042175501508488456261773, −3.35270688127939944859521606360, −2.61930875908172074648985185154, −1.12106752524639717251524380318, −0.63793014732322263644745477697, 0.63793014732322263644745477697, 1.12106752524639717251524380318, 2.61930875908172074648985185154, 3.35270688127939944859521606360, 4.14767042175501508488456261773, 5.06407135204977400579289229628, 6.05564317611295830637091442587, 6.30763414854500950073895718061, 7.26118850252987331100961143908, 7.57092483065320991340095813349

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