Properties

Label 2-825-1.1-c5-0-60
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  − 3.33·2-s + 9·3-s − 20.8·4-s − 30.0·6-s + 103.·7-s + 176.·8-s + 81·9-s + 121·11-s − 187.·12-s − 357.·13-s − 345.·14-s + 78.7·16-s + 2.06e3·17-s − 270.·18-s + 1.32e3·19-s + 931.·21-s − 403.·22-s − 2.49e3·23-s + 1.58e3·24-s + 1.19e3·26-s + 729·27-s − 2.15e3·28-s + 1.65e3·29-s + 4.69e3·31-s − 5.90e3·32-s + 1.08e3·33-s − 6.90e3·34-s + ⋯
L(s)  = 1  − 0.589·2-s + 0.577·3-s − 0.651·4-s − 0.340·6-s + 0.798·7-s + 0.974·8-s + 0.333·9-s + 0.301·11-s − 0.376·12-s − 0.587·13-s − 0.470·14-s + 0.0769·16-s + 1.73·17-s − 0.196·18-s + 0.838·19-s + 0.460·21-s − 0.177·22-s − 0.984·23-s + 0.562·24-s + 0.346·26-s + 0.192·27-s − 0.520·28-s + 0.366·29-s + 0.876·31-s − 1.01·32-s + 0.174·33-s − 1.02·34-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

\(L(3)\) \(\approx\) \(2.168680016\)
\(L(\frac12)\) \(\approx\) \(2.168680016\)
\(L(\frac{7}{2})\) 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 \( 1 + 3.33T + 32T^{2} \)
7 \( 1 - 103.T + 1.68e4T^{2} \)
13 \( 1 + 357.T + 3.71e5T^{2} \)
17 \( 1 - 2.06e3T + 1.41e6T^{2} \)
19 \( 1 - 1.32e3T + 2.47e6T^{2} \)
23 \( 1 + 2.49e3T + 6.43e6T^{2} \)
29 \( 1 - 1.65e3T + 2.05e7T^{2} \)
31 \( 1 - 4.69e3T + 2.86e7T^{2} \)
37 \( 1 - 5.21e3T + 6.93e7T^{2} \)
41 \( 1 - 1.35e4T + 1.15e8T^{2} \)
43 \( 1 - 1.07e4T + 1.47e8T^{2} \)
47 \( 1 + 2.92e3T + 2.29e8T^{2} \)
53 \( 1 + 2.15e4T + 4.18e8T^{2} \)
59 \( 1 + 4.54e3T + 7.14e8T^{2} \)
61 \( 1 - 3.64e3T + 8.44e8T^{2} \)
67 \( 1 + 5.54e4T + 1.35e9T^{2} \)
71 \( 1 - 4.49e3T + 1.80e9T^{2} \)
73 \( 1 - 3.35e4T + 2.07e9T^{2} \)
79 \( 1 + 7.91e3T + 3.07e9T^{2} \)
83 \( 1 + 2.05e4T + 3.93e9T^{2} \)
89 \( 1 + 1.11e5T + 5.58e9T^{2} \)
97 \( 1 + 5.82e4T + 8.58e9T^{2} \)
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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

−9.660919068603323869972036899762, −8.557779528567587504814695676225, −7.83773449774469824351068272731, −7.46608644315081240445528778698, −5.90843792353154713285006805235, −4.90281949418899243826719479876, −4.10673702799699721959935964041, −2.95562605983844381788354165623, −1.59460467851108882611484848166, −0.78062203901852461632972930591, 0.78062203901852461632972930591, 1.59460467851108882611484848166, 2.95562605983844381788354165623, 4.10673702799699721959935964041, 4.90281949418899243826719479876, 5.90843792353154713285006805235, 7.46608644315081240445528778698, 7.83773449774469824351068272731, 8.557779528567587504814695676225, 9.660919068603323869972036899762

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