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Properties

Label 2-825-5.4-c3-0-14
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  + 2.37i·2-s + 3i·3-s + 2.37·4-s − 7.11·6-s − 6.74i·7-s + 24.6i·8-s − 9·9-s + 11·11-s + 7.11i·12-s − 60.9i·13-s + 16·14-s − 39.3·16-s + 99.1i·17-s − 21.3i·18-s − 24.7·19-s + ⋯
L(s)  = 1  + 0.838i·2-s + 0.577i·3-s + 0.296·4-s − 0.484·6-s − 0.364i·7-s + 1.08i·8-s − 0.333·9-s + 0.301·11-s + 0.171i·12-s − 1.30i·13-s + 0.305·14-s − 0.615·16-s + 1.41i·17-s − 0.279i·18-s − 0.298·19-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

\(L(2)\) \(\approx\) \(1.360606326\)
\(L(\frac12)\) \(\approx\) \(1.360606326\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.37iT - 8T^{2} \)
7 \( 1 + 6.74iT - 343T^{2} \)
13 \( 1 + 60.9iT - 2.19e3T^{2} \)
17 \( 1 - 99.1iT - 4.91e3T^{2} \)
19 \( 1 + 24.7T + 6.85e3T^{2} \)
23 \( 1 - 112iT - 1.21e4T^{2} \)
29 \( 1 - 21.1T + 2.43e4T^{2} \)
31 \( 1 + 318.T + 2.97e4T^{2} \)
37 \( 1 - 150. iT - 5.06e4T^{2} \)
41 \( 1 + 252.T + 6.89e4T^{2} \)
43 \( 1 - 214. iT - 7.95e4T^{2} \)
47 \( 1 + 105. iT - 1.03e5T^{2} \)
53 \( 1 - 325. iT - 1.48e5T^{2} \)
59 \( 1 + 196T + 2.05e5T^{2} \)
61 \( 1 + 402.T + 2.26e5T^{2} \)
67 \( 1 + 27.4iT - 3.00e5T^{2} \)
71 \( 1 + 300.T + 3.57e5T^{2} \)
73 \( 1 - 427. iT - 3.89e5T^{2} \)
79 \( 1 + 97.5T + 4.93e5T^{2} \)
83 \( 1 - 1.10e3iT - 5.71e5T^{2} \)
89 \( 1 + 463.T + 7.04e5T^{2} \)
97 \( 1 - 1.79e3iT - 9.12e5T^{2} \)
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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

−10.51069218048522274370083039418, −9.387817002890803888705064865858, −8.387385147995717552187812263212, −7.79324491563703216432106805028, −6.86711223117972757708025309324, −5.88261950766662318032930445976, −5.32405005264839667475787472954, −4.04499045967386209324665594763, −3.06197791381191993155869170195, −1.58808755750868779990321346023, 0.32642839023806344541901065787, 1.70100711505853940977267372537, 2.41796061896203910475026284751, 3.53697824011665292713893105777, 4.67801349241768296738920585915, 5.97447999526733709883579303045, 6.88689209894152707896240179699, 7.37226340761461926845788807258, 8.821847393769693316692582952882, 9.300003973692651938508729375445

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