Properties

Label 2-825-1.1-c5-0-104
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.29·2-s + 9·3-s − 30.3·4-s − 11.6·6-s − 91.7·7-s + 80.5·8-s + 81·9-s − 121·11-s − 272.·12-s − 70.3·13-s + 118.·14-s + 866.·16-s + 1.17e3·17-s − 104.·18-s − 961.·19-s − 825.·21-s + 156.·22-s − 1.30e3·23-s + 725.·24-s + 90.9·26-s + 729·27-s + 2.78e3·28-s − 3.34e3·29-s + 6.45e3·31-s − 3.69e3·32-s − 1.08e3·33-s − 1.51e3·34-s + ⋯
L(s)  = 1  − 0.228·2-s + 0.577·3-s − 0.947·4-s − 0.131·6-s − 0.707·7-s + 0.445·8-s + 0.333·9-s − 0.301·11-s − 0.547·12-s − 0.115·13-s + 0.161·14-s + 0.846·16-s + 0.986·17-s − 0.0761·18-s − 0.611·19-s − 0.408·21-s + 0.0688·22-s − 0.513·23-s + 0.256·24-s + 0.0263·26-s + 0.192·27-s + 0.670·28-s − 0.738·29-s + 1.20·31-s − 0.638·32-s − 0.174·33-s − 0.225·34-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: \(1\)
Selberg data: \((2,\ 825,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 1.29T + 32T^{2} \)
7 \( 1 + 91.7T + 1.68e4T^{2} \)
13 \( 1 + 70.3T + 3.71e5T^{2} \)
17 \( 1 - 1.17e3T + 1.41e6T^{2} \)
19 \( 1 + 961.T + 2.47e6T^{2} \)
23 \( 1 + 1.30e3T + 6.43e6T^{2} \)
29 \( 1 + 3.34e3T + 2.05e7T^{2} \)
31 \( 1 - 6.45e3T + 2.86e7T^{2} \)
37 \( 1 - 1.32e4T + 6.93e7T^{2} \)
41 \( 1 - 1.84e4T + 1.15e8T^{2} \)
43 \( 1 + 1.92e4T + 1.47e8T^{2} \)
47 \( 1 + 1.70e4T + 2.29e8T^{2} \)
53 \( 1 - 3.12e4T + 4.18e8T^{2} \)
59 \( 1 - 4.34e4T + 7.14e8T^{2} \)
61 \( 1 + 3.56e3T + 8.44e8T^{2} \)
67 \( 1 + 7.07e4T + 1.35e9T^{2} \)
71 \( 1 - 4.12e4T + 1.80e9T^{2} \)
73 \( 1 - 4.12e4T + 2.07e9T^{2} \)
79 \( 1 + 5.74e4T + 3.07e9T^{2} \)
83 \( 1 - 1.72e4T + 3.93e9T^{2} \)
89 \( 1 + 2.44e4T + 5.58e9T^{2} \)
97 \( 1 - 1.26e5T + 8.58e9T^{2} \)
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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

−9.113839621630092198534064451531, −8.205758738075863944764179166909, −7.67877261519328944334053849029, −6.46340665107338007937231687945, −5.47299840196202253947483283833, −4.39549070887992927836202087446, −3.57635738773756887209810161052, −2.54322095085225543478526645127, −1.11367988958456383454735177962, 0, 1.11367988958456383454735177962, 2.54322095085225543478526645127, 3.57635738773756887209810161052, 4.39549070887992927836202087446, 5.47299840196202253947483283833, 6.46340665107338007937231687945, 7.67877261519328944334053849029, 8.205758738075863944764179166909, 9.113839621630092198534064451531

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