Properties

Label 2-825-5.4-c1-0-17
Degree $2$
Conductor $825$
Sign $0.894 - 0.447i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.70i·2-s + i·3-s − 5.34·4-s − 2.70·6-s − 1.07i·7-s − 9.04i·8-s − 9-s + 11-s − 5.34i·12-s − 4.34i·13-s + 2.92·14-s + 13.8·16-s − 7.75i·17-s − 2.70i·18-s − 5.26·19-s + ⋯
L(s)  = 1  + 1.91i·2-s + 0.577i·3-s − 2.67·4-s − 1.10·6-s − 0.407i·7-s − 3.19i·8-s − 0.333·9-s + 0.301·11-s − 1.54i·12-s − 1.20i·13-s + 0.780·14-s + 3.45·16-s − 1.88i·17-s − 0.638i·18-s − 1.20·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.628363 + 0.148336i\)
\(L(\frac12)\) \(\approx\) \(0.628363 + 0.148336i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 \)
11 \( 1 - T \)
good2 \( 1 - 2.70iT - 2T^{2} \)
7 \( 1 + 1.07iT - 7T^{2} \)
13 \( 1 + 4.34iT - 13T^{2} \)
17 \( 1 + 7.75iT - 17T^{2} \)
19 \( 1 + 5.26T + 19T^{2} \)
23 \( 1 + 2.15iT - 23T^{2} \)
29 \( 1 + 1.41T + 29T^{2} \)
31 \( 1 + 4.68T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 9.41T + 41T^{2} \)
43 \( 1 - 7.60iT - 43T^{2} \)
47 \( 1 + 4.68iT - 47T^{2} \)
53 \( 1 - 0.156iT - 53T^{2} \)
59 \( 1 + 6.15T + 59T^{2} \)
61 \( 1 + 4.15T + 61T^{2} \)
67 \( 1 - 8.68iT - 67T^{2} \)
71 \( 1 + 4.68T + 71T^{2} \)
73 \( 1 + 10.4iT - 73T^{2} \)
79 \( 1 - 8.09T + 79T^{2} \)
83 \( 1 + 11.0iT - 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + 14.6iT - 97T^{2} \)
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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

−9.953060812244913585966964161523, −9.167763216076040545869024698971, −8.456098331235089754872739872306, −7.60100892860067725123871754128, −6.87884950973636085503960710412, −5.96036151646217843109011494085, −5.08179408184155358246922501546, −4.42055673436722627067979442307, −3.26364735823302301800308967992, −0.32592891858079128549973764519, 1.61313959608575948932764737755, 2.15449222198825069803858356654, 3.58844764605204203956616777997, 4.26505844248580017466678776321, 5.52707032904482241445914892365, 6.58481070371980395265923008600, 8.065116575860255601453218217379, 8.825772603540461271510421507281, 9.353033321869806272516238829174, 10.51809633589231397078135498072

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