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Properties

Label 2-825-55.9-c1-0-5
Degree $2$
Conductor $825$
Sign $0.0615 - 0.998i$
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  + (−1.04 − 0.338i)2-s + (−0.587 − 0.809i)3-s + (−0.647 − 0.470i)4-s + (0.338 + 1.04i)6-s + (0.414 − 0.570i)7-s + (1.80 + 2.48i)8-s + (−0.309 + 0.951i)9-s + (3.31 − 0.189i)11-s + 0.800i·12-s + (−4.48 − 1.45i)13-s + (−0.624 + 0.453i)14-s + (−0.543 − 1.67i)16-s + (−7.39 + 2.40i)17-s + (0.643 − 0.886i)18-s + (−0.970 + 0.705i)19-s + ⋯
L(s)  = 1  + (−0.736 − 0.239i)2-s + (−0.339 − 0.467i)3-s + (−0.323 − 0.235i)4-s + (0.138 + 0.425i)6-s + (0.156 − 0.215i)7-s + (0.637 + 0.877i)8-s + (−0.103 + 0.317i)9-s + (0.998 − 0.0572i)11-s + 0.231i·12-s + (−1.24 − 0.403i)13-s + (−0.166 + 0.121i)14-s + (−0.135 − 0.418i)16-s + (−1.79 + 0.583i)17-s + (0.151 − 0.208i)18-s + (−0.222 + 0.161i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0615 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0615 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.0615 - 0.998i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (724, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.0615 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.170533 + 0.160335i\)
\(L(\frac12)\) \(\approx\) \(0.170533 + 0.160335i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.587 + 0.809i)T \)
5 \( 1 \)
11 \( 1 + (-3.31 + 0.189i)T \)
good2 \( 1 + (1.04 + 0.338i)T + (1.61 + 1.17i)T^{2} \)
7 \( 1 + (-0.414 + 0.570i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (4.48 + 1.45i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (7.39 - 2.40i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.970 - 0.705i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 6.89iT - 23T^{2} \)
29 \( 1 + (-1.07 - 0.780i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.37 - 7.30i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.95 - 6.82i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-0.188 + 0.136i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 7.32iT - 43T^{2} \)
47 \( 1 + (-4.89 - 6.73i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-6.48 - 2.10i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.86 + 2.08i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.35 + 10.3i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 2.04iT - 67T^{2} \)
71 \( 1 + (-0.207 - 0.637i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.94 - 4.04i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.704 - 2.16i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.00 + 0.652i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 3.34T + 89T^{2} \)
97 \( 1 + (3.15 + 1.02i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.60257524174476632141446259674, −9.522421401776998383421369872784, −8.777311976175764043736785226946, −8.095934018307250113648047963324, −6.98812281053287529256358833682, −6.29241929226083224320442632255, −4.97571100917741492650720476420, −4.32031153894354264659790002211, −2.48335961338409462603578299295, −1.31833710079039054545346857799, 0.16527297098398481639552267914, 2.11356017996825423125125068210, 3.84646085029479330979291888423, 4.50271444863628565682962583746, 5.57132474467521487612121782726, 6.93363303424067488919920909288, 7.30028112720974397722964079024, 8.669140548134488718766848616252, 9.177790973681908271160934436389, 9.718574413114931474895626313957

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