Properties

Label 2-825-1.1-c1-0-8
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 $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.90·2-s − 3-s + 1.62·4-s + 1.90·6-s + 4.42·7-s + 0.719·8-s + 9-s + 11-s − 1.62·12-s + 0.622·13-s − 8.42·14-s − 4.61·16-s + 5.18·17-s − 1.90·18-s + 7.05·19-s − 4.42·21-s − 1.90·22-s − 8.85·23-s − 0.719·24-s − 1.18·26-s − 27-s + 7.18·28-s − 7.80·29-s + 2.75·31-s + 7.34·32-s − 33-s − 9.86·34-s + ⋯
L(s)  = 1  − 1.34·2-s − 0.577·3-s + 0.811·4-s + 0.776·6-s + 1.67·7-s + 0.254·8-s + 0.333·9-s + 0.301·11-s − 0.468·12-s + 0.172·13-s − 2.25·14-s − 1.15·16-s + 1.25·17-s − 0.448·18-s + 1.61·19-s − 0.966·21-s − 0.405·22-s − 1.84·23-s − 0.146·24-s − 0.232·26-s − 0.192·27-s + 1.35·28-s − 1.44·29-s + 0.494·31-s + 1.29·32-s − 0.174·33-s − 1.69·34-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: $\chi_{825} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8169526999\)
\(L(\frac12)\) \(\approx\) \(0.8169526999\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.90T + 2T^{2} \)
7 \( 1 - 4.42T + 7T^{2} \)
13 \( 1 - 0.622T + 13T^{2} \)
17 \( 1 - 5.18T + 17T^{2} \)
19 \( 1 - 7.05T + 19T^{2} \)
23 \( 1 + 8.85T + 23T^{2} \)
29 \( 1 + 7.80T + 29T^{2} \)
31 \( 1 - 2.75T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 0.193T + 41T^{2} \)
43 \( 1 + 5.67T + 43T^{2} \)
47 \( 1 - 2.75T + 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 + 4.85T + 59T^{2} \)
61 \( 1 - 6.85T + 61T^{2} \)
67 \( 1 - 1.24T + 67T^{2} \)
71 \( 1 - 2.75T + 71T^{2} \)
73 \( 1 + 4.23T + 73T^{2} \)
79 \( 1 - 8.56T + 79T^{2} \)
83 \( 1 + 0.133T + 83T^{2} \)
89 \( 1 - 5.61T + 89T^{2} \)
97 \( 1 + 7.24T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.09655303064597242717470941601, −9.509729547902967316931157586557, −8.409578853511103745454259503108, −7.80148911629081451534797706316, −7.26252456607141353211515782971, −5.82175593387592209920033989527, −5.04109463830870922650412143532, −3.90153366285414162564750922574, −1.91027492340035911089610934674, −1.01429119232244115145294490384, 1.01429119232244115145294490384, 1.91027492340035911089610934674, 3.90153366285414162564750922574, 5.04109463830870922650412143532, 5.82175593387592209920033989527, 7.26252456607141353211515782971, 7.80148911629081451534797706316, 8.409578853511103745454259503108, 9.509729547902967316931157586557, 10.09655303064597242717470941601

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