Properties

Label 2-825-165.47-c1-0-11
Degree $2$
Conductor $825$
Sign $-0.225 + 0.974i$
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  + (−0.321 + 2.03i)2-s + (−0.0270 + 1.73i)3-s + (−2.12 − 0.689i)4-s + (−3.51 − 0.612i)6-s + (2.94 − 1.49i)7-s + (0.216 − 0.425i)8-s + (−2.99 − 0.0937i)9-s + (−3.09 + 1.18i)11-s + (1.25 − 3.65i)12-s + (−0.948 + 5.98i)13-s + (2.09 + 6.46i)14-s + (−2.81 − 2.04i)16-s + (0.201 − 0.0319i)17-s + (1.15 − 6.06i)18-s + (−0.0382 + 0.0124i)19-s + ⋯
L(s)  = 1  + (−0.227 + 1.43i)2-s + (−0.0156 + 0.999i)3-s + (−1.06 − 0.344i)4-s + (−1.43 − 0.249i)6-s + (1.11 − 0.566i)7-s + (0.0766 − 0.150i)8-s + (−0.999 − 0.0312i)9-s + (−0.933 + 0.357i)11-s + (0.361 − 1.05i)12-s + (−0.262 + 1.66i)13-s + (0.561 + 1.72i)14-s + (−0.704 − 0.511i)16-s + (0.0488 − 0.00774i)17-s + (0.272 − 1.42i)18-s + (−0.00878 + 0.00285i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.225 + 0.974i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.225 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.534929 - 0.672725i\)
\(L(\frac12)\) \(\approx\) \(0.534929 - 0.672725i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0270 - 1.73i)T \)
5 \( 1 \)
11 \( 1 + (3.09 - 1.18i)T \)
good2 \( 1 + (0.321 - 2.03i)T + (-1.90 - 0.618i)T^{2} \)
7 \( 1 + (-2.94 + 1.49i)T + (4.11 - 5.66i)T^{2} \)
13 \( 1 + (0.948 - 5.98i)T + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (-0.201 + 0.0319i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (0.0382 - 0.0124i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (5.24 + 5.24i)T + 23iT^{2} \)
29 \( 1 + (0.634 - 1.95i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.70 - 4.14i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.13 - 1.59i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (-2.14 + 0.697i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (3.63 - 3.63i)T - 43iT^{2} \)
47 \( 1 + (-7.08 - 3.60i)T + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (-8.88 - 1.40i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (-0.225 + 0.693i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.70 + 3.41i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-9.28 - 9.28i)T + 67iT^{2} \)
71 \( 1 + (-5.39 + 7.43i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.57 - 8.98i)T + (-42.9 + 59.0i)T^{2} \)
79 \( 1 + (0.190 + 0.261i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.317 - 2.00i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + 17.6T + 89T^{2} \)
97 \( 1 + (-4.11 - 0.651i)T + (92.2 + 29.9i)T^{2} \)
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

−10.69044119519127616658509279338, −9.791597517767728781646390652344, −8.888601542126817882804125321855, −8.254615782017185650839090151595, −7.43868183036908806363191489640, −6.61135804411944467995361747446, −5.47714987049203178471889957673, −4.77095992818971751039072555674, −4.10516544853113163952680197325, −2.25629014262108169108437831496, 0.43288739921168049819481740353, 1.83661526314151439908177037886, 2.55570627373587170182646037899, 3.61629532559990646154344504916, 5.27666685833189928898998571343, 5.79658085607425786541179994239, 7.41689575639255344010548140274, 8.057108160772384292867530613571, 8.728542117826696111958065529147, 9.876706673920056474807041324499

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