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Properties

Label 2-825-11.3-c1-0-36
Degree $2$
Conductor $825$
Sign $-0.394 + 0.918i$
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  + (2.06 − 1.49i)2-s + (0.309 + 0.951i)3-s + (1.39 − 4.28i)4-s + (2.06 + 1.49i)6-s + (1.44 − 4.45i)7-s + (−1.97 − 6.09i)8-s + (−0.809 + 0.587i)9-s + (−2.54 − 2.12i)11-s + 4.51·12-s + (−2.89 + 2.10i)13-s + (−3.69 − 11.3i)14-s + (−5.92 − 4.30i)16-s + (1.02 + 0.741i)17-s + (−0.788 + 2.42i)18-s + (2.18 + 6.71i)19-s + ⋯
L(s)  = 1  + (1.45 − 1.06i)2-s + (0.178 + 0.549i)3-s + (0.696 − 2.14i)4-s + (0.842 + 0.612i)6-s + (0.546 − 1.68i)7-s + (−0.699 − 2.15i)8-s + (−0.269 + 0.195i)9-s + (−0.767 − 0.641i)11-s + 1.30·12-s + (−0.802 + 0.583i)13-s + (−0.986 − 3.03i)14-s + (−1.48 − 1.07i)16-s + (0.247 + 0.179i)17-s + (−0.185 + 0.571i)18-s + (0.500 + 1.54i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.394 + 0.918i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.394 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96711 - 2.98575i\)
\(L(\frac12)\) \(\approx\) \(1.96711 - 2.98575i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 \)
11 \( 1 + (2.54 + 2.12i)T \)
good2 \( 1 + (-2.06 + 1.49i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (-1.44 + 4.45i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.89 - 2.10i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.02 - 0.741i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.18 - 6.71i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.08T + 23T^{2} \)
29 \( 1 + (-1.91 + 5.89i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.0537 + 0.0390i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.22 - 3.76i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.62 - 4.99i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 9.77T + 43T^{2} \)
47 \( 1 + (-1.67 - 5.14i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-10.6 + 7.73i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.163 - 0.504i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.96 + 3.60i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 3.10T + 67T^{2} \)
71 \( 1 + (6.73 + 4.89i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (5.15 - 15.8i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.10 - 4.43i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.90 - 2.83i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 6.05T + 89T^{2} \)
97 \( 1 + (2.46 - 1.78i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.19263148441652013579170267987, −9.795563582664646071608927322196, −8.166936941790772380694287532326, −7.30134431506059343314829401534, −6.02191989988917938816207639683, −5.09013800819168683785116099170, −4.30145580170787524558322527961, −3.68734174455607948897651815635, −2.61461284062223799337413220518, −1.18172891194111928982139578493, 2.44923188731380357378814177831, 2.92833375736992886444108879329, 4.65624627767906692533883878429, 5.30243425989170602074767360989, 5.81439811280273294111660844220, 7.17110806295462215082306810796, 7.42711857848070311709757064866, 8.537507550940299132562811703560, 9.207249036333172858807331846070, 10.78948724293300551954534589049

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