Properties

Label 2-9075-1.1-c1-0-102
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  + 1.73·2-s − 3-s + 0.999·4-s − 1.73·6-s + 2·7-s − 1.73·8-s + 9-s − 0.999·12-s − 1.46·13-s + 3.46·14-s − 5·16-s + 1.73·18-s + 1.46·19-s − 2·21-s + 6.92·23-s + 1.73·24-s − 2.53·26-s − 27-s + 1.99·28-s − 3.46·29-s + 2.92·31-s − 5.19·32-s + 0.999·36-s − 8.92·37-s + 2.53·38-s + 1.46·39-s + 3.46·41-s + ⋯
L(s)  = 1  + 1.22·2-s − 0.577·3-s + 0.499·4-s − 0.707·6-s + 0.755·7-s − 0.612·8-s + 0.333·9-s − 0.288·12-s − 0.406·13-s + 0.925·14-s − 1.25·16-s + 0.408·18-s + 0.335·19-s − 0.436·21-s + 1.44·23-s + 0.353·24-s − 0.497·26-s − 0.192·27-s + 0.377·28-s − 0.643·29-s + 0.525·31-s − 0.918·32-s + 0.166·36-s − 1.46·37-s + 0.411·38-s + 0.234·39-s + 0.541·41-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

\(L(1)\) \(\approx\) \(3.042843751\)
\(L(\frac12)\) \(\approx\) \(3.042843751\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T + 2T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
13 \( 1 + 1.46T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 1.46T + 19T^{2} \)
23 \( 1 - 6.92T + 23T^{2} \)
29 \( 1 + 3.46T + 29T^{2} \)
31 \( 1 - 2.92T + 31T^{2} \)
37 \( 1 + 8.92T + 37T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 - 8.92T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 - 12.9T + 53T^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 + 12.9T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−7.42256212662911566491520518172, −6.93311411081517571091800658187, −6.13434784022473980558383703390, −5.40688228906277810701054417927, −5.02187513565407239809682955277, −4.42241689248316417824866728046, −3.64266446915300404072296382123, −2.84353079120802432503798271730, −1.90631496592735315835785895063, −0.71960825319775932200572858032, 0.71960825319775932200572858032, 1.90631496592735315835785895063, 2.84353079120802432503798271730, 3.64266446915300404072296382123, 4.42241689248316417824866728046, 5.02187513565407239809682955277, 5.40688228906277810701054417927, 6.13434784022473980558383703390, 6.93311411081517571091800658187, 7.42256212662911566491520518172

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