↑

Properties

Label 2-825-275.246-c1-0-53
Degree $2$
Conductor $825$
Sign $-0.991 - 0.131i$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.363i)2-s + (0.809 − 0.587i)3-s + (−0.5 + 1.53i)4-s + (−0.690 − 2.12i)5-s + (−0.190 + 0.587i)6-s + (−0.927 − 2.85i)7-s + (−0.690 − 2.12i)8-s + (0.309 − 0.951i)9-s + (1.11 + 0.812i)10-s + (−2.19 + 2.48i)11-s + (0.499 + 1.53i)12-s + (−0.381 + 1.17i)13-s + (1.50 + 1.08i)14-s + (−1.80 − 1.31i)15-s + (−1.49 − 1.08i)16-s + (−1.61 + 4.97i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.256i)2-s + (0.467 − 0.339i)3-s + (−0.250 + 0.769i)4-s + (−0.309 − 0.951i)5-s + (−0.0779 + 0.239i)6-s + (−0.350 − 1.07i)7-s + (−0.244 − 0.751i)8-s + (0.103 − 0.317i)9-s + (0.353 + 0.256i)10-s + (−0.660 + 0.750i)11-s + (0.144 + 0.444i)12-s + (−0.105 + 0.326i)13-s + (0.400 + 0.291i)14-s + (−0.467 − 0.339i)15-s + (−0.374 − 0.272i)16-s + (−0.392 + 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.991 - 0.131i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (796, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.991 - 0.131i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (0.690 + 2.12i)T \)
11 \( 1 + (2.19 - 2.48i)T \)
good2 \( 1 + (0.5 - 0.363i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (0.927 + 2.85i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (0.381 - 1.17i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.61 - 4.97i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.11 + 3.44i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.5 - 4.61i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (3.19 - 9.82i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + 5.76T + 31T^{2} \)
37 \( 1 + (-1.73 + 1.26i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + (-0.881 + 0.640i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.47 + 10.6i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-11.7 + 8.50i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.45 + 4.47i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-2.42 + 1.76i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 + (1.54 + 4.75i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.35 - 7.24i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-0.690 + 2.12i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-3.54 - 10.9i)T + (-78.4 + 57.0i)T^{2} \)
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

−9.498621752398567670117802887924, −8.832879997174867776845051935815, −8.074975771452653296949727510827, −7.32123161334087670419230565092, −6.80366401123376073657738098065, −5.17097538236043121501927414858, −4.11617667248104771668021516535, −3.47953928792298006864853723238, −1.74930780554351980452551852782, 0, 2.31819851287167964098382944437, 2.89385954104433795474517428164, 4.27836152520680742291100880135, 5.58858926336970791521682943407, 6.08393998742236711450600402396, 7.40413127521042526104119186891, 8.376525934969546335284584536396, 9.033151612600208838458749762167, 9.941205194795642473428186153182

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