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Properties

Label 2-825-5.4-c3-0-21
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  + 5.06i·2-s + 3i·3-s − 17.6·4-s − 15.1·6-s − 27.4i·7-s − 48.8i·8-s − 9·9-s + 11·11-s − 52.9i·12-s + 22.6i·13-s + 138.·14-s + 106.·16-s + 41.1i·17-s − 45.5i·18-s + 142.·19-s + ⋯
L(s)  = 1  + 1.79i·2-s + 0.577i·3-s − 2.20·4-s − 1.03·6-s − 1.48i·7-s − 2.16i·8-s − 0.333·9-s + 0.301·11-s − 1.27i·12-s + 0.484i·13-s + 2.65·14-s + 1.66·16-s + 0.587i·17-s − 0.596i·18-s + 1.71·19-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

\(L(2)\) \(\approx\) \(1.437150675\)
\(L(\frac12)\) \(\approx\) \(1.437150675\)
\(L(\frac{5}{2})\) 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 \( 1 - 5.06iT - 8T^{2} \)
7 \( 1 + 27.4iT - 343T^{2} \)
13 \( 1 - 22.6iT - 2.19e3T^{2} \)
17 \( 1 - 41.1iT - 4.91e3T^{2} \)
19 \( 1 - 142.T + 6.85e3T^{2} \)
23 \( 1 - 176. iT - 1.21e4T^{2} \)
29 \( 1 + 76.2T + 2.43e4T^{2} \)
31 \( 1 - 197.T + 2.97e4T^{2} \)
37 \( 1 + 367. iT - 5.06e4T^{2} \)
41 \( 1 + 238.T + 6.89e4T^{2} \)
43 \( 1 - 30.2iT - 7.95e4T^{2} \)
47 \( 1 + 137. iT - 1.03e5T^{2} \)
53 \( 1 - 638. iT - 1.48e5T^{2} \)
59 \( 1 + 103.T + 2.05e5T^{2} \)
61 \( 1 - 605.T + 2.26e5T^{2} \)
67 \( 1 - 704. iT - 3.00e5T^{2} \)
71 \( 1 + 782.T + 3.57e5T^{2} \)
73 \( 1 + 243. iT - 3.89e5T^{2} \)
79 \( 1 - 532.T + 4.93e5T^{2} \)
83 \( 1 - 1.20e3iT - 5.71e5T^{2} \)
89 \( 1 + 1.05e3T + 7.04e5T^{2} \)
97 \( 1 - 85.1iT - 9.12e5T^{2} \)
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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

−9.872145869101620407335916907586, −9.417535299439748829412158176037, −8.406637490699615891266652482416, −7.42959776785435568551175626829, −7.13603476452005474007454908997, −5.98507927757299402377190110491, −5.20292942119828082152017479754, −4.20015006078409455046818128372, −3.59320612105945100773166326472, −1.03882447543936260588392575268, 0.47687039029893101136449582455, 1.61863219666894656830595526358, 2.68744476850135246790732910097, 3.19113403497388354539600519829, 4.72900577533175402130504185335, 5.47829446415968041462854949434, 6.65576724987017303629081226371, 8.114072306238404646224681177646, 8.724703520598441471530212813960, 9.565923070704405831789205419484

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