Properties

Label 2-825-1.1-c3-0-38
Degree $2$
Conductor $825$
Sign $1$
Analytic cond. $48.6765$
Root an. cond. $6.97686$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3·3-s − 7·4-s + 3·6-s + 26·7-s − 15·8-s + 9·9-s + 11·11-s − 21·12-s + 32·13-s + 26·14-s + 41·16-s − 74·17-s + 9·18-s − 60·19-s + 78·21-s + 11·22-s + 182·23-s − 45·24-s + 32·26-s + 27·27-s − 182·28-s − 90·29-s − 8·31-s + 161·32-s + 33·33-s − 74·34-s + ⋯
L(s)  = 1  + 0.353·2-s + 0.577·3-s − 7/8·4-s + 0.204·6-s + 1.40·7-s − 0.662·8-s + 1/3·9-s + 0.301·11-s − 0.505·12-s + 0.682·13-s + 0.496·14-s + 0.640·16-s − 1.05·17-s + 0.117·18-s − 0.724·19-s + 0.810·21-s + 0.106·22-s + 1.64·23-s − 0.382·24-s + 0.241·26-s + 0.192·27-s − 1.22·28-s − 0.576·29-s − 0.0463·31-s + 0.889·32-s + 0.174·33-s − 0.373·34-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.982211532\)
\(L(\frac12)\) \(\approx\) \(2.982211532\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - p T \)
5 \( 1 \)
11 \( 1 - p T \)
good2 \( 1 - T + p^{3} T^{2} \)
7 \( 1 - 26 T + p^{3} T^{2} \)
13 \( 1 - 32 T + p^{3} T^{2} \)
17 \( 1 + 74 T + p^{3} T^{2} \)
19 \( 1 + 60 T + p^{3} T^{2} \)
23 \( 1 - 182 T + p^{3} T^{2} \)
29 \( 1 + 90 T + p^{3} T^{2} \)
31 \( 1 + 8 T + p^{3} T^{2} \)
37 \( 1 - 66 T + p^{3} T^{2} \)
41 \( 1 - 422 T + p^{3} T^{2} \)
43 \( 1 + 408 T + p^{3} T^{2} \)
47 \( 1 - 506 T + p^{3} T^{2} \)
53 \( 1 + 348 T + p^{3} T^{2} \)
59 \( 1 + 200 T + p^{3} T^{2} \)
61 \( 1 - 132 T + p^{3} T^{2} \)
67 \( 1 - 1036 T + p^{3} T^{2} \)
71 \( 1 - 762 T + p^{3} T^{2} \)
73 \( 1 - 542 T + p^{3} T^{2} \)
79 \( 1 + 550 T + p^{3} T^{2} \)
83 \( 1 - 132 T + p^{3} T^{2} \)
89 \( 1 - 570 T + p^{3} T^{2} \)
97 \( 1 + 14 T + p^{3} T^{2} \)
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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

−9.583345757957409493349995630346, −8.793165414019917646668046376778, −8.417628688043093599676955650668, −7.42047833014835507355357172359, −6.26430233038168183949653502064, −5.08760325235589785539595274130, −4.45317900765150053239862040574, −3.58325798115085777687166212781, −2.20059608316522608129176369566, −0.941720156224116459450854943026, 0.941720156224116459450854943026, 2.20059608316522608129176369566, 3.58325798115085777687166212781, 4.45317900765150053239862040574, 5.08760325235589785539595274130, 6.26430233038168183949653502064, 7.42047833014835507355357172359, 8.417628688043093599676955650668, 8.793165414019917646668046376778, 9.583345757957409493349995630346

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