Properties

Label 2-825-33.32-c1-0-38
Degree $2$
Conductor $825$
Sign $-0.288 + 0.957i$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.65 − 0.5i)3-s − 2·4-s + (2.5 + 1.65i)9-s + 3.31i·11-s + (3.31 + i)12-s + 4·16-s − 9i·23-s + (−3.31 − 4i)27-s − 5·31-s + (1.65 − 5.5i)33-s + (−5 − 3.31i)36-s − 9.94·37-s − 6.63i·44-s − 12i·47-s + (−6.63 − 2i)48-s + 7·49-s + ⋯
L(s)  = 1  + (−0.957 − 0.288i)3-s − 4-s + (0.833 + 0.552i)9-s + 1.00i·11-s + (0.957 + 0.288i)12-s + 16-s − 1.87i·23-s + (−0.638 − 0.769i)27-s − 0.898·31-s + (0.288 − 0.957i)33-s + (−0.833 − 0.552i)36-s − 1.63·37-s − 1.00i·44-s − 1.75i·47-s + (−0.957 − 0.288i)48-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.288 + 0.957i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (626, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.288 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.295428 - 0.397640i\)
\(L(\frac12)\) \(\approx\) \(0.295428 - 0.397640i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.65 + 0.5i)T \)
5 \( 1 \)
11 \( 1 - 3.31iT \)
good2 \( 1 + 2T^{2} \)
7 \( 1 - 7T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 9iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 + 9.94T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 12iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 3.31iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 9.94T + 67T^{2} \)
71 \( 1 + 16.5iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 16.5iT - 89T^{2} \)
97 \( 1 - 9.94T + 97T^{2} \)
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

−10.20068315501162388722551590827, −9.167684860479123016631325485559, −8.310372534203210591290919028599, −7.27880225648591179336223759098, −6.50782503277666358392566002653, −5.35031418448083963065659489179, −4.75782028154387791004990478000, −3.80737018210525945412835870765, −1.96420001903979055192800385570, −0.33467337291877393707617747332, 1.18082830095439656368622583428, 3.39934128202570735807086305435, 4.18076235095347934769139061450, 5.39624003169409678417574263387, 5.69128670001264019738298771088, 6.97113932821447455354628354830, 7.971673836944407701905124461845, 9.021897832432353117219259421708, 9.579265784015767060084363521840, 10.53050532450580465989296668673

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