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Properties

Label 2-825-5.4-c3-0-70
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  + 0.793i·2-s + 3i·3-s + 7.37·4-s − 2.38·6-s − 2.90i·7-s + 12.1i·8-s − 9·9-s + 11·11-s + 22.1i·12-s − 68.4i·13-s + 2.30·14-s + 49.2·16-s − 31.0i·17-s − 7.14i·18-s − 54.9·19-s + ⋯
L(s)  = 1  + 0.280i·2-s + 0.577i·3-s + 0.921·4-s − 0.161·6-s − 0.157i·7-s + 0.539i·8-s − 0.333·9-s + 0.301·11-s + 0.531i·12-s − 1.46i·13-s + 0.0440·14-s + 0.770·16-s − 0.443i·17-s − 0.0935i·18-s − 0.663·19-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

\(L(2)\) \(\approx\) \(2.318640130\)
\(L(\frac12)\) \(\approx\) \(2.318640130\)
\(L(\frac{5}{2})\) 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 \( 1 - 0.793iT - 8T^{2} \)
7 \( 1 + 2.90iT - 343T^{2} \)
13 \( 1 + 68.4iT - 2.19e3T^{2} \)
17 \( 1 + 31.0iT - 4.91e3T^{2} \)
19 \( 1 + 54.9T + 6.85e3T^{2} \)
23 \( 1 + 180. iT - 1.21e4T^{2} \)
29 \( 1 + 67.3T + 2.43e4T^{2} \)
31 \( 1 - 153.T + 2.97e4T^{2} \)
37 \( 1 + 324. iT - 5.06e4T^{2} \)
41 \( 1 + 25.4T + 6.89e4T^{2} \)
43 \( 1 + 133. iT - 7.95e4T^{2} \)
47 \( 1 - 113. iT - 1.03e5T^{2} \)
53 \( 1 + 91.6iT - 1.48e5T^{2} \)
59 \( 1 + 434.T + 2.05e5T^{2} \)
61 \( 1 + 60.2T + 2.26e5T^{2} \)
67 \( 1 + 439. iT - 3.00e5T^{2} \)
71 \( 1 - 436.T + 3.57e5T^{2} \)
73 \( 1 - 91.5iT - 3.89e5T^{2} \)
79 \( 1 + 947.T + 4.93e5T^{2} \)
83 \( 1 - 944. iT - 5.71e5T^{2} \)
89 \( 1 + 413.T + 7.04e5T^{2} \)
97 \( 1 + 1.46e3iT - 9.12e5T^{2} \)
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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

−9.988966801953675233892891939049, −8.805227027775087308494283530718, −8.057435427573461277144664519825, −7.17575011533614250607053596533, −6.23773432416399713115456879014, −5.48500386409653542013556854523, −4.38940842471443480560323974266, −3.18983046544914945612514976341, −2.28453076832656237795547445711, −0.59301688577943080213199833952, 1.32895516038687583035557829936, 2.05218779474017054605741575589, 3.23165999077668277307095940491, 4.34405164345601302467216594663, 5.78154596573569885365521366206, 6.53046940646653797871426993057, 7.17604689174648089775974134162, 8.100928343876885979354068232865, 9.089144729964741040787762549557, 9.945829935745694744915262633980

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