Properties

Label 2-825-1.1-c5-0-110
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  − 5.72·2-s − 9·3-s + 0.820·4-s + 51.5·6-s + 158.·7-s + 178.·8-s + 81·9-s + 121·11-s − 7.38·12-s + 78.2·13-s − 906.·14-s − 1.04e3·16-s − 88.2·17-s − 464.·18-s + 1.86e3·19-s − 1.42e3·21-s − 693.·22-s + 503.·23-s − 1.60e3·24-s − 448.·26-s − 729·27-s + 129.·28-s + 1.05e3·29-s − 9.05e3·31-s + 297.·32-s − 1.08e3·33-s + 505.·34-s + ⋯
L(s)  = 1  − 1.01·2-s − 0.577·3-s + 0.0256·4-s + 0.584·6-s + 1.22·7-s + 0.986·8-s + 0.333·9-s + 0.301·11-s − 0.0148·12-s + 0.128·13-s − 1.23·14-s − 1.02·16-s − 0.0740·17-s − 0.337·18-s + 1.18·19-s − 0.704·21-s − 0.305·22-s + 0.198·23-s − 0.569·24-s − 0.130·26-s − 0.192·27-s + 0.0312·28-s + 0.231·29-s − 1.69·31-s + 0.0512·32-s − 0.174·33-s + 0.0749·34-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 \( 1 + 5.72T + 32T^{2} \)
7 \( 1 - 158.T + 1.68e4T^{2} \)
13 \( 1 - 78.2T + 3.71e5T^{2} \)
17 \( 1 + 88.2T + 1.41e6T^{2} \)
19 \( 1 - 1.86e3T + 2.47e6T^{2} \)
23 \( 1 - 503.T + 6.43e6T^{2} \)
29 \( 1 - 1.05e3T + 2.05e7T^{2} \)
31 \( 1 + 9.05e3T + 2.86e7T^{2} \)
37 \( 1 + 3.89e3T + 6.93e7T^{2} \)
41 \( 1 + 9.45e3T + 1.15e8T^{2} \)
43 \( 1 + 8.91e3T + 1.47e8T^{2} \)
47 \( 1 + 9.43e3T + 2.29e8T^{2} \)
53 \( 1 + 2.55e4T + 4.18e8T^{2} \)
59 \( 1 - 9.90e3T + 7.14e8T^{2} \)
61 \( 1 + 1.56e4T + 8.44e8T^{2} \)
67 \( 1 - 1.68e4T + 1.35e9T^{2} \)
71 \( 1 - 7.14e4T + 1.80e9T^{2} \)
73 \( 1 + 6.57e4T + 2.07e9T^{2} \)
79 \( 1 - 1.04e5T + 3.07e9T^{2} \)
83 \( 1 - 1.54e4T + 3.93e9T^{2} \)
89 \( 1 - 2.77e3T + 5.58e9T^{2} \)
97 \( 1 + 1.25e5T + 8.58e9T^{2} \)
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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

−9.092859093189143789832013865935, −8.216939276609400254333254930964, −7.56722978482392281977331739095, −6.70605540723172968229124398183, −5.32698778784365579220206874217, −4.81450305982020334209908882314, −3.62329235228260096523457224709, −1.82505589295614982906236005993, −1.16191170564195881975332305457, 0, 1.16191170564195881975332305457, 1.82505589295614982906236005993, 3.62329235228260096523457224709, 4.81450305982020334209908882314, 5.32698778784365579220206874217, 6.70605540723172968229124398183, 7.56722978482392281977331739095, 8.216939276609400254333254930964, 9.092859093189143789832013865935

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