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Properties

Label 2-825-5.4-c3-0-61
Degree $2$
Conductor $825$
Sign $0.447 + 0.894i$
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  − 0.723i·2-s − 3i·3-s + 7.47·4-s − 2.17·6-s + 1.13i·7-s − 11.1i·8-s − 9·9-s + 11·11-s − 22.4i·12-s + 21.8i·13-s + 0.819·14-s + 51.7·16-s + 6.18i·17-s + 6.51i·18-s + 92.8·19-s + ⋯
L(s)  = 1  − 0.255i·2-s − 0.577i·3-s + 0.934·4-s − 0.147·6-s + 0.0611i·7-s − 0.494i·8-s − 0.333·9-s + 0.301·11-s − 0.539i·12-s + 0.466i·13-s + 0.0156·14-s + 0.807·16-s + 0.0881i·17-s + 0.0852i·18-s + 1.12·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.447 + 0.894i$
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: \((2,\ 825,\ (\ :3/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.835305324\)
\(L(\frac12)\) \(\approx\) \(2.835305324\)
\(L(\frac{5}{2})\) 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 \( 1 + 0.723iT - 8T^{2} \)
7 \( 1 - 1.13iT - 343T^{2} \)
13 \( 1 - 21.8iT - 2.19e3T^{2} \)
17 \( 1 - 6.18iT - 4.91e3T^{2} \)
19 \( 1 - 92.8T + 6.85e3T^{2} \)
23 \( 1 - 36.7iT - 1.21e4T^{2} \)
29 \( 1 + 71.1T + 2.43e4T^{2} \)
31 \( 1 - 186.T + 2.97e4T^{2} \)
37 \( 1 + 356. iT - 5.06e4T^{2} \)
41 \( 1 - 271.T + 6.89e4T^{2} \)
43 \( 1 + 155. iT - 7.95e4T^{2} \)
47 \( 1 - 234. iT - 1.03e5T^{2} \)
53 \( 1 + 195. iT - 1.48e5T^{2} \)
59 \( 1 - 455.T + 2.05e5T^{2} \)
61 \( 1 + 441.T + 2.26e5T^{2} \)
67 \( 1 - 133. iT - 3.00e5T^{2} \)
71 \( 1 + 1.04e3T + 3.57e5T^{2} \)
73 \( 1 - 160. iT - 3.89e5T^{2} \)
79 \( 1 - 761.T + 4.93e5T^{2} \)
83 \( 1 + 51.7iT - 5.71e5T^{2} \)
89 \( 1 - 1.07e3T + 7.04e5T^{2} \)
97 \( 1 + 703. iT - 9.12e5T^{2} \)
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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

−9.724220338447943312730640894544, −8.890582211207758290657345324510, −7.69082224480297003584656115505, −7.20371219552598323755119712820, −6.26546789261145547364935550635, −5.50315680733399533611012004446, −4.02480459051119246771829423014, −2.92570262279310880952907456440, −1.93704146998362575344802419368, −0.879310619942989634262106537295, 1.08107589526650009528952651053, 2.56374454620790454541480667019, 3.43460880633942993475535888588, 4.70347844369199870351734376640, 5.66859490803345064863169945120, 6.46287477924686266761874348105, 7.43517486229607141450492795005, 8.164807367343194393443286988131, 9.195676431059487898776081356638, 10.10738891155090986074697587622

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