Properties

Label 2-825-1.1-c5-0-32
Degree $2$
Conductor $825$
Sign $1$
Analytic cond. $132.316$
Root an. cond. $11.5028$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.31·2-s + 9·3-s − 30.2·4-s + 11.8·6-s − 87.4·7-s − 82.0·8-s + 81·9-s + 121·11-s − 272.·12-s + 521.·13-s − 115.·14-s + 860.·16-s − 68.1·17-s + 106.·18-s − 1.58e3·19-s − 787.·21-s + 159.·22-s + 191.·23-s − 738.·24-s + 686.·26-s + 729·27-s + 2.64e3·28-s − 4.25e3·29-s − 83.6·31-s + 3.75e3·32-s + 1.08e3·33-s − 89.7·34-s + ⋯
L(s)  = 1  + 0.232·2-s + 0.577·3-s − 0.945·4-s + 0.134·6-s − 0.674·7-s − 0.453·8-s + 0.333·9-s + 0.301·11-s − 0.546·12-s + 0.855·13-s − 0.157·14-s + 0.840·16-s − 0.0571·17-s + 0.0776·18-s − 1.00·19-s − 0.389·21-s + 0.0702·22-s + 0.0756·23-s − 0.261·24-s + 0.199·26-s + 0.192·27-s + 0.638·28-s − 0.939·29-s − 0.0156·31-s + 0.648·32-s + 0.174·33-s − 0.0133·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(132.316\)
Root analytic conductor: \(11.5028\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.817269247\)
\(L(\frac12)\) \(\approx\) \(1.817269247\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 9T \)
5 \( 1 \)
11 \( 1 - 121T \)
good2 \( 1 - 1.31T + 32T^{2} \)
7 \( 1 + 87.4T + 1.68e4T^{2} \)
13 \( 1 - 521.T + 3.71e5T^{2} \)
17 \( 1 + 68.1T + 1.41e6T^{2} \)
19 \( 1 + 1.58e3T + 2.47e6T^{2} \)
23 \( 1 - 191.T + 6.43e6T^{2} \)
29 \( 1 + 4.25e3T + 2.05e7T^{2} \)
31 \( 1 + 83.6T + 2.86e7T^{2} \)
37 \( 1 + 1.44e4T + 6.93e7T^{2} \)
41 \( 1 - 6.15e3T + 1.15e8T^{2} \)
43 \( 1 - 1.40e3T + 1.47e8T^{2} \)
47 \( 1 + 1.47e4T + 2.29e8T^{2} \)
53 \( 1 - 1.09e4T + 4.18e8T^{2} \)
59 \( 1 + 1.60e4T + 7.14e8T^{2} \)
61 \( 1 - 4.10e4T + 8.44e8T^{2} \)
67 \( 1 + 1.25e3T + 1.35e9T^{2} \)
71 \( 1 - 4.76e4T + 1.80e9T^{2} \)
73 \( 1 + 7.35e3T + 2.07e9T^{2} \)
79 \( 1 - 9.01e4T + 3.07e9T^{2} \)
83 \( 1 - 1.08e4T + 3.93e9T^{2} \)
89 \( 1 - 1.12e5T + 5.58e9T^{2} \)
97 \( 1 - 9.69e4T + 8.58e9T^{2} \)
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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

−9.309376843858148673002023187446, −8.749927628168522095535822378272, −8.008454274463396117673558250351, −6.80602941513035923220187516440, −5.98406029273359512951296313040, −4.90755088258900252997806835162, −3.84617802883444688107782419284, −3.35620195450311527974443322409, −1.92800345538882350535471115198, −0.57237827489796841002637860026, 0.57237827489796841002637860026, 1.92800345538882350535471115198, 3.35620195450311527974443322409, 3.84617802883444688107782419284, 4.90755088258900252997806835162, 5.98406029273359512951296313040, 6.80602941513035923220187516440, 8.008454274463396117673558250351, 8.749927628168522095535822378272, 9.309376843858148673002023187446

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