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Properties

Label 2-825-11.5-c1-0-17
Degree $2$
Conductor $825$
Sign $0.634 - 0.772i$
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  + (0.392 + 1.20i)2-s + (−0.809 − 0.587i)3-s + (0.311 − 0.226i)4-s + (0.392 − 1.20i)6-s + (−0.390 + 0.284i)7-s + (2.45 + 1.78i)8-s + (0.309 + 0.951i)9-s + (−0.982 − 3.16i)11-s − 0.384·12-s + (0.971 + 2.99i)13-s + (−0.496 − 0.361i)14-s + (−0.952 + 2.93i)16-s + (−0.775 + 2.38i)17-s + (−1.02 + 0.747i)18-s + (3.00 + 2.18i)19-s + ⋯
L(s)  = 1  + (0.277 + 0.854i)2-s + (−0.467 − 0.339i)3-s + (0.155 − 0.113i)4-s + (0.160 − 0.493i)6-s + (−0.147 + 0.107i)7-s + (0.866 + 0.629i)8-s + (0.103 + 0.317i)9-s + (−0.296 − 0.955i)11-s − 0.111·12-s + (0.269 + 0.829i)13-s + (−0.132 − 0.0964i)14-s + (−0.238 + 0.732i)16-s + (−0.188 + 0.578i)17-s + (−0.242 + 0.176i)18-s + (0.690 + 0.501i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 - 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.634 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.634 - 0.772i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (676, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.634 - 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64796 + 0.779059i\)
\(L(\frac12)\) \(\approx\) \(1.64796 + 0.779059i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 \)
11 \( 1 + (0.982 + 3.16i)T \)
good2 \( 1 + (-0.392 - 1.20i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + (0.390 - 0.284i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.971 - 2.99i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.775 - 2.38i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.00 - 2.18i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 8.53T + 23T^{2} \)
29 \( 1 + (-8.07 + 5.86i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.78 + 5.50i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.59 - 1.88i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.36 - 4.62i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 5.97T + 43T^{2} \)
47 \( 1 + (-4.96 - 3.60i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.488 - 1.50i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.305 + 0.222i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.929 + 2.86i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 5.98T + 67T^{2} \)
71 \( 1 + (-2.57 + 7.93i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.122 + 0.0889i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.31 - 7.11i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.25 + 3.85i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 17.0T + 89T^{2} \)
97 \( 1 + (1.11 + 3.43i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.55584700088945633922537154584, −9.409866880232040943785886277427, −8.350921829980655596859205860686, −7.63933405316790799692913560234, −6.63444286825327119916771930781, −6.13105492262702106198892871570, −5.31028214019973325726287037660, −4.31383592529091578872082483655, −2.77941937835163091781031311239, −1.27141286439250328579353724594, 1.08955816111322313245081328138, 2.67149825737576848838643037653, 3.45223523544761934921617866136, 4.71368735898962627655235515505, 5.29846654781782780417272289231, 6.93809830552465897861289442143, 7.16958655340606026817340854501, 8.582799008407923556793125126682, 9.578404965181297101957988536632, 10.45081058651424266528133952312

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