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Properties

Label 2-825-5.4-c3-0-29
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  − 2.56i·2-s − 3i·3-s + 1.43·4-s − 7.68·6-s + 6.24i·7-s − 24.1i·8-s − 9·9-s − 11·11-s − 4.31i·12-s + 49.1i·13-s + 16·14-s − 50.4·16-s + 82.7i·17-s + 23.0i·18-s + 130.·19-s + ⋯
L(s)  = 1  − 0.905i·2-s − 0.577i·3-s + 0.179·4-s − 0.522·6-s + 0.337i·7-s − 1.06i·8-s − 0.333·9-s − 0.301·11-s − 0.103i·12-s + 1.04i·13-s + 0.305·14-s − 0.787·16-s + 1.17i·17-s + 0.301i·18-s + 1.57·19-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

\(L(2)\) \(\approx\) \(2.086093858\)
\(L(\frac12)\) \(\approx\) \(2.086093858\)
\(L(\frac{5}{2})\) 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 \( 1 + 2.56iT - 8T^{2} \)
7 \( 1 - 6.24iT - 343T^{2} \)
13 \( 1 - 49.1iT - 2.19e3T^{2} \)
17 \( 1 - 82.7iT - 4.91e3T^{2} \)
19 \( 1 - 130.T + 6.85e3T^{2} \)
23 \( 1 - 185. iT - 1.21e4T^{2} \)
29 \( 1 - 8.90T + 2.43e4T^{2} \)
31 \( 1 - 5.26T + 2.97e4T^{2} \)
37 \( 1 + 416. iT - 5.06e4T^{2} \)
41 \( 1 + 298.T + 6.89e4T^{2} \)
43 \( 1 - 513. iT - 7.95e4T^{2} \)
47 \( 1 - 557. iT - 1.03e5T^{2} \)
53 \( 1 - 168. iT - 1.48e5T^{2} \)
59 \( 1 + 618.T + 2.05e5T^{2} \)
61 \( 1 - 786.T + 2.26e5T^{2} \)
67 \( 1 + 339. iT - 3.00e5T^{2} \)
71 \( 1 - 1.12e3T + 3.57e5T^{2} \)
73 \( 1 - 123. iT - 3.89e5T^{2} \)
79 \( 1 - 309.T + 4.93e5T^{2} \)
83 \( 1 - 1.02e3iT - 5.71e5T^{2} \)
89 \( 1 - 141.T + 7.04e5T^{2} \)
97 \( 1 - 798. iT - 9.12e5T^{2} \)
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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

−9.704685718377023441611621619988, −9.273982315396676629066825462647, −7.966005549766058679501940758975, −7.25389990429451598427013067060, −6.31839091473566131092179525575, −5.42287925055142412297937772806, −3.98251697883205898659105989023, −3.06225517693603393875307386883, −1.95780278737256209974146234013, −1.17681491253224488590353459131, 0.58196040904336874519970965885, 2.49037797879548003224072618429, 3.43155501171034160453977048251, 5.00997838876306786949897323657, 5.29828134623063684670295875590, 6.54128650735784058590856815824, 7.26409122590379874320780763425, 8.109004567893520568525678062617, 8.821563197584928950875157829731, 10.05054208086214427461482657548

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