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Properties

Label 2-825-165.74-c1-0-21
Degree $2$
Conductor $825$
Sign $-0.399 + 0.916i$
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  + (−1.11 − 1.53i)2-s + (−1.08 − 1.34i)3-s + (−0.499 + 1.53i)4-s + (−0.857 + 3.18i)6-s + (−0.726 + 2.23i)7-s + (−0.690 + 0.224i)8-s + (−0.633 + 2.93i)9-s + (3.07 + 1.23i)11-s + (2.61 − 1.00i)12-s + (4.25 − 1.38i)14-s + (3.73 + 2.71i)16-s + (3.61 − 4.97i)17-s + (5.22 − 2.30i)18-s + (−4.30 + 1.40i)19-s + (3.80 − 1.45i)21-s + (−1.53 − 6.11i)22-s + ⋯
L(s)  = 1  + (−0.790 − 1.08i)2-s + (−0.628 − 0.778i)3-s + (−0.249 + 0.769i)4-s + (−0.350 + 1.29i)6-s + (−0.274 + 0.845i)7-s + (−0.244 + 0.0793i)8-s + (−0.211 + 0.977i)9-s + (0.927 + 0.372i)11-s + (0.755 − 0.288i)12-s + (1.13 − 0.369i)14-s + (0.934 + 0.678i)16-s + (0.877 − 1.20i)17-s + (1.23 − 0.542i)18-s + (−0.988 + 0.321i)19-s + (0.830 − 0.317i)21-s + (−0.328 − 1.30i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 + 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.399 + 0.916i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.399 + 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.401455 - 0.612991i\)
\(L(\frac12)\) \(\approx\) \(0.401455 - 0.612991i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.08 + 1.34i)T \)
5 \( 1 \)
11 \( 1 + (-3.07 - 1.23i)T \)
good2 \( 1 + (1.11 + 1.53i)T + (-0.618 + 1.90i)T^{2} \)
7 \( 1 + (0.726 - 2.23i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.61 + 4.97i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (4.30 - 1.40i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.09T + 23T^{2} \)
29 \( 1 + (-2.21 + 6.80i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.04 + 2.93i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-4.47 - 1.45i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.224 - 0.690i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.33T + 43T^{2} \)
47 \( 1 + (-1.57 - 4.84i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (6.16 - 4.47i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (5.20 + 1.69i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.89 + 6.74i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 12.3iT - 67T^{2} \)
71 \( 1 + (-6.88 + 9.47i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.224 + 0.690i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.54 - 2.12i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-9.47 + 13.0i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 0.854iT - 89T^{2} \)
97 \( 1 + (9.99 + 13.7i)T + (-29.9 + 92.2i)T^{2} \)
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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

−9.883581797380262040142824995763, −9.408372577139169817385598928129, −8.398967218468631715156158089972, −7.60609010031697107839725291244, −6.32415282636058261380543853716, −5.84646100045568534613838600335, −4.43186265543105211955841392853, −2.85446538703019720881404529081, −2.02057985079295589178046743979, −0.75252599809667592406427452794, 0.887420331552426145148292726806, 3.42425869179160856088766438137, 4.21481762960191833977580455815, 5.52121711769749357279634148815, 6.38701407328701123400038451401, 6.83502244101250882096405913719, 8.021657875766729907825199031535, 8.777311446740340806492508542531, 9.544924970070880518513593887788, 10.35002843611270043868566271665

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