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Properties

Label 2-825-825.263-c0-0-1
Degree $2$
Conductor $825$
Sign $0.904 + 0.425i$
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  + (−0.587 + 0.809i)3-s + (−0.951 + 0.309i)4-s − i·5-s + (−0.309 − 0.951i)9-s + (0.587 − 0.809i)11-s + (0.309 − 0.951i)12-s + (0.809 + 0.587i)15-s + (0.809 − 0.587i)16-s + (0.309 + 0.951i)20-s + (1.76 − 0.278i)23-s − 25-s + (0.951 + 0.309i)27-s + (0.363 − 1.11i)31-s + (0.309 + 0.951i)33-s + (0.587 + 0.809i)36-s + (−0.221 + 1.39i)37-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)3-s + (−0.951 + 0.309i)4-s − i·5-s + (−0.309 − 0.951i)9-s + (0.587 − 0.809i)11-s + (0.309 − 0.951i)12-s + (0.809 + 0.587i)15-s + (0.809 − 0.587i)16-s + (0.309 + 0.951i)20-s + (1.76 − 0.278i)23-s − 25-s + (0.951 + 0.309i)27-s + (0.363 − 1.11i)31-s + (0.309 + 0.951i)33-s + (0.587 + 0.809i)36-s + (−0.221 + 1.39i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.904 + 0.425i$
Analytic conductor: \(0.411728\)
Root analytic conductor: \(0.641660\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :0),\ 0.904 + 0.425i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6407464620\)
\(L(\frac12)\) \(\approx\) \(0.6407464620\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.587 - 0.809i)T \)
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

−10.14200027994532303388808845162, −9.490274137521920394968551642858, −8.737458580020904945104185544205, −8.288864778458434202392819801786, −6.78049535518620670703092691389, −5.62706301101465045440716640712, −4.97702283511995855540013803148, −4.17770020904118249243858176508, −3.29695462264750076196912513136, −0.844164821024954791460282420236, 1.42837860351992838937221460310, 2.93056573868422154250291997691, 4.30343148230735570622569330537, 5.26510630523877096916566314909, 6.21134620104732134265887177993, 7.02802881686710212735298055842, 7.70261346737980020472718022976, 8.891699955843179358224665708912, 9.648926339305918454021457931494, 10.68594966624517103032899133311

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