Properties

Label 2-825-1.1-c1-0-20
Degree $2$
Conductor $825$
Sign $-1$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.67·2-s + 3-s + 5.15·4-s − 2.67·6-s − 2.80·7-s − 8.44·8-s + 9-s − 11-s + 5.15·12-s + 5.11·13-s + 7.50·14-s + 12.2·16-s − 4.54·17-s − 2.67·18-s − 4.57·19-s − 2.80·21-s + 2.67·22-s − 4·23-s − 8.44·24-s − 13.6·26-s + 27-s − 14.4·28-s − 2.38·29-s − 0.962·31-s − 15.9·32-s − 33-s + 12.1·34-s + ⋯
L(s)  = 1  − 1.89·2-s + 0.577·3-s + 2.57·4-s − 1.09·6-s − 1.06·7-s − 2.98·8-s + 0.333·9-s − 0.301·11-s + 1.48·12-s + 1.41·13-s + 2.00·14-s + 3.06·16-s − 1.10·17-s − 0.630·18-s − 1.04·19-s − 0.612·21-s + 0.570·22-s − 0.834·23-s − 1.72·24-s − 2.68·26-s + 0.192·27-s − 2.73·28-s − 0.443·29-s − 0.172·31-s − 2.81·32-s − 0.174·33-s + 2.08·34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 + 2.67T + 2T^{2} \)
7 \( 1 + 2.80T + 7T^{2} \)
13 \( 1 - 5.11T + 13T^{2} \)
17 \( 1 + 4.54T + 17T^{2} \)
19 \( 1 + 4.57T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 2.38T + 29T^{2} \)
31 \( 1 + 0.962T + 31T^{2} \)
37 \( 1 + 1.61T + 37T^{2} \)
41 \( 1 + 2.38T + 41T^{2} \)
43 \( 1 + 2.80T + 43T^{2} \)
47 \( 1 - 4.31T + 47T^{2} \)
53 \( 1 - 6.57T + 53T^{2} \)
59 \( 1 + 13.2T + 59T^{2} \)
61 \( 1 - 7.92T + 61T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 + 7.35T + 71T^{2} \)
73 \( 1 + 6.41T + 73T^{2} \)
79 \( 1 - 1.35T + 79T^{2} \)
83 \( 1 - 0.806T + 83T^{2} \)
89 \( 1 + 2.96T + 89T^{2} \)
97 \( 1 - 9.92T + 97T^{2} \)
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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

−9.642618645744077532626415376066, −8.827688934420347286038269668205, −8.492362227673343736911577944408, −7.48239538220345664434792632423, −6.58698742743786968829934548755, −6.03644935278392690009694102950, −3.90239618656781203444884810085, −2.77787129063554322030528680232, −1.70933572110942068536528085561, 0, 1.70933572110942068536528085561, 2.77787129063554322030528680232, 3.90239618656781203444884810085, 6.03644935278392690009694102950, 6.58698742743786968829934548755, 7.48239538220345664434792632423, 8.492362227673343736911577944408, 8.827688934420347286038269668205, 9.642618645744077532626415376066

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