Properties

Label 2-825-15.8-c1-0-56
Degree $2$
Conductor $825$
Sign $-0.0618 + 0.998i$
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  + (0.292 + 0.292i)2-s + (1 − 1.41i)3-s − 1.82i·4-s + (0.707 − 0.121i)6-s + (3.41 − 3.41i)7-s + (1.12 − 1.12i)8-s + (−1.00 − 2.82i)9-s + i·11-s + (−2.58 − 1.82i)12-s + (2 + 2i)13-s + 2·14-s − 3·16-s + (2.82 + 2.82i)17-s + (0.535 − 1.12i)18-s + 2.82i·19-s + ⋯
L(s)  = 1  + (0.207 + 0.207i)2-s + (0.577 − 0.816i)3-s − 0.914i·4-s + (0.288 − 0.0495i)6-s + (1.29 − 1.29i)7-s + (0.396 − 0.396i)8-s + (−0.333 − 0.942i)9-s + 0.301i·11-s + (−0.746 − 0.527i)12-s + (0.554 + 0.554i)13-s + 0.534·14-s − 0.750·16-s + (0.685 + 0.685i)17-s + (0.126 − 0.264i)18-s + 0.648i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0618 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0618 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.0618 + 0.998i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (518, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.0618 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64951 - 1.75491i\)
\(L(\frac12)\) \(\approx\) \(1.64951 - 1.75491i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1 + 1.41i)T \)
5 \( 1 \)
11 \( 1 - iT \)
good2 \( 1 + (-0.292 - 0.292i)T + 2iT^{2} \)
7 \( 1 + (-3.41 + 3.41i)T - 7iT^{2} \)
13 \( 1 + (-2 - 2i)T + 13iT^{2} \)
17 \( 1 + (-2.82 - 2.82i)T + 17iT^{2} \)
19 \( 1 - 2.82iT - 19T^{2} \)
23 \( 1 + (3.24 - 3.24i)T - 23iT^{2} \)
29 \( 1 - 0.828T + 29T^{2} \)
31 \( 1 + 3.17T + 31T^{2} \)
37 \( 1 + (-0.171 + 0.171i)T - 37iT^{2} \)
41 \( 1 - 7.65iT - 41T^{2} \)
43 \( 1 + (0.242 + 0.242i)T + 43iT^{2} \)
47 \( 1 + (-7.24 - 7.24i)T + 47iT^{2} \)
53 \( 1 + (1 - i)T - 53iT^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 6.48T + 61T^{2} \)
67 \( 1 + (-6.41 + 6.41i)T - 67iT^{2} \)
71 \( 1 - 2.48iT - 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 10.8iT - 79T^{2} \)
83 \( 1 + (-9.07 + 9.07i)T - 83iT^{2} \)
89 \( 1 + 9.65T + 89T^{2} \)
97 \( 1 + (10.6 - 10.6i)T - 97iT^{2} \)
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

−10.05496846521669166470346520616, −9.122994622941190182470648958315, −7.963969650053418577878602877789, −7.57427848200681111029076161372, −6.56917759832328570109786603654, −5.74765079580935279076186878179, −4.53084510211319525980807390011, −3.71475029611568090445096209226, −1.79107502245595639152077496611, −1.21848365658770454487868104945, 2.15576498815130194013354878143, 2.95070071559490373775013961281, 4.01765575517228638209779848461, 5.00163778593237984102344560267, 5.66173687138756236383748585362, 7.34894779745338604898395644848, 8.261922167908599565847723082897, 8.573381332142190227491678965824, 9.378813667905948599610294844731, 10.63438156848232709138481486253

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