Properties

Label 2-825-5.4-c1-0-20
Degree $2$
Conductor $825$
Sign $-0.447 + 0.894i$
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  − 1.36i·2-s + i·3-s + 0.141·4-s + 1.36·6-s − 2.50i·7-s − 2.91i·8-s − 9-s + 11-s + 0.141i·12-s + 1.14i·13-s − 3.41·14-s − 3.69·16-s − 7.64i·17-s + 1.36i·18-s − 1.77·19-s + ⋯
L(s)  = 1  − 0.964i·2-s + 0.577i·3-s + 0.0706·4-s + 0.556·6-s − 0.946i·7-s − 1.03i·8-s − 0.333·9-s + 0.301·11-s + 0.0408i·12-s + 0.316i·13-s − 0.912·14-s − 0.924·16-s − 1.85i·17-s + 0.321i·18-s − 0.407·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.447 + 0.894i$
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: \((2,\ 825,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.840395 - 1.35978i\)
\(L(\frac12)\) \(\approx\) \(0.840395 - 1.35978i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.36iT - 2T^{2} \)
7 \( 1 + 2.50iT - 7T^{2} \)
13 \( 1 - 1.14iT - 13T^{2} \)
17 \( 1 + 7.64iT - 17T^{2} \)
19 \( 1 + 1.77T + 19T^{2} \)
23 \( 1 - 1.41iT - 23T^{2} \)
29 \( 1 - 0.726T + 29T^{2} \)
31 \( 1 - 2.85T + 31T^{2} \)
37 \( 1 - 8.42iT - 37T^{2} \)
41 \( 1 - 0.636T + 41T^{2} \)
43 \( 1 + 12.6iT - 43T^{2} \)
47 \( 1 + 6.14iT - 47T^{2} \)
53 \( 1 + 12.0iT - 53T^{2} \)
59 \( 1 - 3.41T + 59T^{2} \)
61 \( 1 - 4.59T + 61T^{2} \)
67 \( 1 - 9.32iT - 67T^{2} \)
71 \( 1 - 5.85T + 71T^{2} \)
73 \( 1 - 7.55iT - 73T^{2} \)
79 \( 1 + 6.91T + 79T^{2} \)
83 \( 1 + 6.17iT - 83T^{2} \)
89 \( 1 + 3.45T + 89T^{2} \)
97 \( 1 - 19.4iT - 97T^{2} \)
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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

−10.06147372378910565847734877043, −9.526755138888262585547647389072, −8.494554817135161229854542717659, −7.17647913187698786037813900902, −6.70949637662139867228051969005, −5.21840213223943142923554578482, −4.21327859517911766577734547427, −3.42886479257348325944268052734, −2.32648292819594252770638668375, −0.789668403743869964662877748110, 1.76338113636392119566882099693, 2.83294617430316222245350672914, 4.40477711544612496636891062686, 5.79122601932226475460322227704, 6.04322377474091933811099576631, 6.96359711452926003331953126943, 8.012379542549033958867909808643, 8.416444927760385272075907488360, 9.332481895112223114183513356203, 10.65004035351097424352994618335

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