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Properties

Label 2-825-5.4-c3-0-65
Degree $2$
Conductor $825$
Sign $0.894 + 0.447i$
Analytic cond. $48.6765$
Root an. cond. $6.97686$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.63i·2-s − 3i·3-s − 5.23·4-s + 10.9·6-s + 20.8i·7-s + 10.0i·8-s − 9·9-s − 11·11-s + 15.7i·12-s − 67.4i·13-s − 75.8·14-s − 78.4·16-s − 57.8i·17-s − 32.7i·18-s + 7.98·19-s + ⋯
L(s)  = 1  + 1.28i·2-s − 0.577i·3-s − 0.654·4-s + 0.742·6-s + 1.12i·7-s + 0.444i·8-s − 0.333·9-s − 0.301·11-s + 0.377i·12-s − 1.44i·13-s − 1.44·14-s − 1.22·16-s − 0.824i·17-s − 0.428i·18-s + 0.0964·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(48.6765\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :3/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.096483851\)
\(L(\frac12)\) \(\approx\) \(1.096483851\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3iT \)
5 \( 1 \)
11 \( 1 + 11T \)
good2 \( 1 - 3.63iT - 8T^{2} \)
7 \( 1 - 20.8iT - 343T^{2} \)
13 \( 1 + 67.4iT - 2.19e3T^{2} \)
17 \( 1 + 57.8iT - 4.91e3T^{2} \)
19 \( 1 - 7.98T + 6.85e3T^{2} \)
23 \( 1 + 67.5iT - 1.21e4T^{2} \)
29 \( 1 - 56.1T + 2.43e4T^{2} \)
31 \( 1 + 127.T + 2.97e4T^{2} \)
37 \( 1 + 95.4iT - 5.06e4T^{2} \)
41 \( 1 + 485.T + 6.89e4T^{2} \)
43 \( 1 - 146. iT - 7.95e4T^{2} \)
47 \( 1 - 164. iT - 1.03e5T^{2} \)
53 \( 1 + 431. iT - 1.48e5T^{2} \)
59 \( 1 - 804.T + 2.05e5T^{2} \)
61 \( 1 + 120.T + 2.26e5T^{2} \)
67 \( 1 + 371. iT - 3.00e5T^{2} \)
71 \( 1 - 529.T + 3.57e5T^{2} \)
73 \( 1 + 1.05e3iT - 3.89e5T^{2} \)
79 \( 1 - 168.T + 4.93e5T^{2} \)
83 \( 1 - 144. iT - 5.71e5T^{2} \)
89 \( 1 - 1.40e3T + 7.04e5T^{2} \)
97 \( 1 - 29.6iT - 9.12e5T^{2} \)
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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

−9.429575290610239243142183089443, −8.501327115753755376012361046784, −8.034224707320285331237810331768, −7.15853771512536241320072496196, −6.33735725663352263304398050077, −5.48646493634845587992712099503, −5.02081882731903522390651264302, −3.10359091744935463562464855148, −2.15710125506939309885933813797, −0.29823562147517371672693777004, 1.17092527358104957904308922266, 2.22279873768538268304823673060, 3.61845376919415731041155666924, 4.00958254684341963832888619613, 5.06415045444656525004754732918, 6.52208895626591537119060260709, 7.27046168668999361401854688749, 8.538840288748405695721448163358, 9.406400285949536477896804451567, 10.19800117261917400836424404954

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