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Properties

Label 2-825-825.527-c0-0-1
Degree $2$
Conductor $825$
Sign $-0.992 + 0.125i$
Analytic cond. $0.411728$
Root an. cond. $0.641660$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)3-s + (−0.951 − 0.309i)4-s − i·5-s + (−0.809 + 0.587i)9-s + (−0.587 − 0.809i)11-s + 0.999i·12-s + (−0.951 + 0.309i)15-s + (0.809 + 0.587i)16-s + (−0.309 + 0.951i)20-s + (−1.76 − 0.278i)23-s − 25-s + (0.809 + 0.587i)27-s + (0.363 + 1.11i)31-s + (−0.587 + 0.809i)33-s + (0.951 − 0.309i)36-s + (−0.221 − 1.39i)37-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)3-s + (−0.951 − 0.309i)4-s − i·5-s + (−0.809 + 0.587i)9-s + (−0.587 − 0.809i)11-s + 0.999i·12-s + (−0.951 + 0.309i)15-s + (0.809 + 0.587i)16-s + (−0.309 + 0.951i)20-s + (−1.76 − 0.278i)23-s − 25-s + (0.809 + 0.587i)27-s + (0.363 + 1.11i)31-s + (−0.587 + 0.809i)33-s + (0.951 − 0.309i)36-s + (−0.221 − 1.39i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.992 + 0.125i$
Analytic conductor: \(0.411728\)
Root analytic conductor: \(0.641660\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :0),\ -0.992 + 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4795175474\)
\(L(\frac12)\) \(\approx\) \(0.4795175474\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + iT \)
11 \( 1 + (0.587 + 0.809i)T \)
good2 \( 1 + (0.951 + 0.309i)T^{2} \)
7 \( 1 - iT^{2} \)
13 \( 1 + (-0.951 + 0.309i)T^{2} \)
17 \( 1 + (-0.587 - 0.809i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (1.76 + 0.278i)T + (0.951 + 0.309i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.221 + 1.39i)T + (-0.951 + 0.309i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.642 + 1.26i)T + (-0.587 + 0.809i)T^{2} \)
53 \( 1 + (-1.76 + 0.896i)T + (0.587 - 0.809i)T^{2} \)
59 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.809 + 1.58i)T + (-0.587 - 0.809i)T^{2} \)
71 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.951 + 0.309i)T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.587 + 0.809i)T^{2} \)
89 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.278 + 0.142i)T + (0.587 - 0.809i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.999988787068063262373544815544, −8.940677903769696787391798573652, −8.339165320279189554666847044986, −7.73943879178065716129610778529, −6.32839715444749050715654116857, −5.55822361481768071714962103257, −4.89782099441090008358061899639, −3.66551165908730171829921417021, −1.93953211410501952469502770082, −0.51276616136996274215340704142, 2.58710438200648431660924031663, 3.72749680691850918128272554673, 4.42980121086140964642187040003, 5.44547427814806156324732161567, 6.34207034023721292147924103846, 7.59709474403710844062221017793, 8.298243689854554743561949277097, 9.470342331301778581229371093891, 9.990980069323676639454475340936, 10.52920802422778068465786180664

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