Properties

Label 2-9075-1.1-c1-0-8
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  − 2.23·2-s − 3-s + 3.00·4-s + 2.23·6-s − 2.23·7-s − 2.23·8-s + 9-s − 3.00·12-s − 4.47·13-s + 5.00·14-s − 0.999·16-s − 2.23·17-s − 2.23·18-s + 2.23·19-s + 2.23·21-s − 23-s + 2.23·24-s + 10.0·26-s − 27-s − 6.70·28-s − 4.47·29-s + 10·31-s + 6.70·32-s + 5.00·34-s + 3.00·36-s − 7·37-s − 5.00·38-s + ⋯
L(s)  = 1  − 1.58·2-s − 0.577·3-s + 1.50·4-s + 0.912·6-s − 0.845·7-s − 0.790·8-s + 0.333·9-s − 0.866·12-s − 1.24·13-s + 1.33·14-s − 0.249·16-s − 0.542·17-s − 0.527·18-s + 0.512·19-s + 0.487·21-s − 0.208·23-s + 0.456·24-s + 1.96·26-s − 0.192·27-s − 1.26·28-s − 0.830·29-s + 1.79·31-s + 1.18·32-s + 0.857·34-s + 0.500·36-s − 1.15·37-s − 0.811·38-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

\(L(1)\) \(\approx\) \(0.2076897556\)
\(L(\frac12)\) \(\approx\) \(0.2076897556\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23T + 2T^{2} \)
7 \( 1 + 2.23T + 7T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 + 2.23T + 17T^{2} \)
19 \( 1 - 2.23T + 19T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + 6.70T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 - 14T + 53T^{2} \)
59 \( 1 + 5T + 59T^{2} \)
61 \( 1 + 4.47T + 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + 13T + 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 - 15.6T + 79T^{2} \)
83 \( 1 + 4.47T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 17T + 97T^{2} \)
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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

−7.76192094165004854603704448548, −7.03768230879161747941393746850, −6.75433199910406747599775043338, −5.92584166136270839402669702447, −5.05901342180944699249273902689, −4.30640883207593010222370834478, −3.15767471252882894590347002319, −2.35579065082866895125588703332, −1.42827924628738642032778457069, −0.29700908034329452138770317144, 0.29700908034329452138770317144, 1.42827924628738642032778457069, 2.35579065082866895125588703332, 3.15767471252882894590347002319, 4.30640883207593010222370834478, 5.05901342180944699249273902689, 5.92584166136270839402669702447, 6.75433199910406747599775043338, 7.03768230879161747941393746850, 7.76192094165004854603704448548

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