Properties

Label 2-825-1.1-c3-0-68
Degree $2$
Conductor $825$
Sign $-1$
Analytic cond. $48.6765$
Root an. cond. $6.97686$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.03·2-s + 3·3-s + 8.31·4-s − 12.1·6-s + 6.49·7-s − 1.27·8-s + 9·9-s + 11·11-s + 24.9·12-s − 30.5·13-s − 26.2·14-s − 61.3·16-s + 43.1·17-s − 36.3·18-s − 6.46·19-s + 19.4·21-s − 44.4·22-s − 108.·23-s − 3.83·24-s + 123.·26-s + 27·27-s + 54.0·28-s − 274.·29-s − 68.5·31-s + 258.·32-s + 33·33-s − 174.·34-s + ⋯
L(s)  = 1  − 1.42·2-s + 0.577·3-s + 1.03·4-s − 0.824·6-s + 0.350·7-s − 0.0564·8-s + 0.333·9-s + 0.301·11-s + 0.600·12-s − 0.651·13-s − 0.501·14-s − 0.958·16-s + 0.615·17-s − 0.476·18-s − 0.0780·19-s + 0.202·21-s − 0.430·22-s − 0.980·23-s − 0.0325·24-s + 0.930·26-s + 0.192·27-s + 0.364·28-s − 1.75·29-s − 0.397·31-s + 1.42·32-s + 0.174·33-s − 0.878·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(48.6765\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 825,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
5 \( 1 \)
11 \( 1 - 11T \)
good2 \( 1 + 4.03T + 8T^{2} \)
7 \( 1 - 6.49T + 343T^{2} \)
13 \( 1 + 30.5T + 2.19e3T^{2} \)
17 \( 1 - 43.1T + 4.91e3T^{2} \)
19 \( 1 + 6.46T + 6.85e3T^{2} \)
23 \( 1 + 108.T + 1.21e4T^{2} \)
29 \( 1 + 274.T + 2.43e4T^{2} \)
31 \( 1 + 68.5T + 2.97e4T^{2} \)
37 \( 1 - 402.T + 5.06e4T^{2} \)
41 \( 1 + 268.T + 6.89e4T^{2} \)
43 \( 1 - 30.5T + 7.95e4T^{2} \)
47 \( 1 + 31.5T + 1.03e5T^{2} \)
53 \( 1 + 252.T + 1.48e5T^{2} \)
59 \( 1 + 558.T + 2.05e5T^{2} \)
61 \( 1 - 335.T + 2.26e5T^{2} \)
67 \( 1 + 28.7T + 3.00e5T^{2} \)
71 \( 1 - 89.0T + 3.57e5T^{2} \)
73 \( 1 - 717.T + 3.89e5T^{2} \)
79 \( 1 - 200.T + 4.93e5T^{2} \)
83 \( 1 + 243.T + 5.71e5T^{2} \)
89 \( 1 + 312.T + 7.04e5T^{2} \)
97 \( 1 + 27.9T + 9.12e5T^{2} \)
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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

−9.574217398620147983242426140936, −8.576179336098556867510594689081, −7.82620663025620707697244771930, −7.36543990800846358327776663057, −6.20075512470464969751454236091, −4.87705170518997450586048041319, −3.72804636049900806135500907917, −2.31405420026820104086321520550, −1.40967291240188729297178882749, 0, 1.40967291240188729297178882749, 2.31405420026820104086321520550, 3.72804636049900806135500907917, 4.87705170518997450586048041319, 6.20075512470464969751454236091, 7.36543990800846358327776663057, 7.82620663025620707697244771930, 8.576179336098556867510594689081, 9.574217398620147983242426140936

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