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Properties

Label 2-825-825.428-c0-0-1
Degree $2$
Conductor $825$
Sign $0.770 + 0.637i$
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  + (0.809 − 0.587i)3-s + (0.587 − 0.809i)4-s + i·5-s + (0.309 − 0.951i)9-s + (−0.951 − 0.309i)11-s − i·12-s + (0.587 + 0.809i)15-s + (−0.309 − 0.951i)16-s + (0.809 + 0.587i)20-s + (0.896 + 1.76i)23-s − 25-s + (−0.309 − 0.951i)27-s + (−1.53 + 1.11i)31-s + (−0.951 + 0.309i)33-s + (−0.587 − 0.809i)36-s + (1.26 + 0.642i)37-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s + (0.587 − 0.809i)4-s + i·5-s + (0.309 − 0.951i)9-s + (−0.951 − 0.309i)11-s − i·12-s + (0.587 + 0.809i)15-s + (−0.309 − 0.951i)16-s + (0.809 + 0.587i)20-s + (0.896 + 1.76i)23-s − 25-s + (−0.309 − 0.951i)27-s + (−1.53 + 1.11i)31-s + (−0.951 + 0.309i)33-s + (−0.587 − 0.809i)36-s + (1.26 + 0.642i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.770 + 0.637i$
Analytic conductor: \(0.411728\)
Root analytic conductor: \(0.641660\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (428, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :0),\ 0.770 + 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.359645236\)
\(L(\frac12)\) \(\approx\) \(1.359645236\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 - iT \)
11 \( 1 + (0.951 + 0.309i)T \)
good2 \( 1 + (-0.587 + 0.809i)T^{2} \)
7 \( 1 + iT^{2} \)
13 \( 1 + (0.587 + 0.809i)T^{2} \)
17 \( 1 + (-0.951 - 0.309i)T^{2} \)
19 \( 1 + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.896 - 1.76i)T + (-0.587 + 0.809i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (-1.26 - 0.642i)T + (0.587 + 0.809i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.221 + 1.39i)T + (-0.951 + 0.309i)T^{2} \)
53 \( 1 + (0.896 - 0.142i)T + (0.951 - 0.309i)T^{2} \)
59 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.309 - 1.95i)T + (-0.951 - 0.309i)T^{2} \)
71 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.587 + 0.809i)T^{2} \)
79 \( 1 + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.951 + 0.309i)T^{2} \)
89 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (-1.76 + 0.278i)T + (0.951 - 0.309i)T^{2} \)
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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

−10.26657988353649189004308478354, −9.644422959695691692352409329735, −8.606296174842911274105140839167, −7.43491212514371568072953418577, −7.14556894341914639129894191381, −6.10899466827752039359948944292, −5.25823698345561762098616928990, −3.48853972864093085692149660726, −2.72543769005277565590877891230, −1.63580873263475397377844462174, 2.08687701078055480369417199867, 3.00385941876144152960033456260, 4.21251451940156494364097363107, 4.87614385524366768937533175071, 6.15446634983255904172094144116, 7.63600956806679753812114580051, 7.84029106228471811457179564866, 8.899058665908273714747207400993, 9.397341573134603665171294555925, 10.60668951350543692884817317498

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