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Properties

Label 2-825-5.4-c3-0-59
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  + 4.03i·2-s + 3i·3-s − 8.31·4-s − 12.1·6-s − 6.49i·7-s − 1.27i·8-s − 9·9-s + 11·11-s − 24.9i·12-s − 30.5i·13-s + 26.2·14-s − 61.3·16-s − 43.1i·17-s − 36.3i·18-s + 6.46·19-s + ⋯
L(s)  = 1  + 1.42i·2-s + 0.577i·3-s − 1.03·4-s − 0.824·6-s − 0.350i·7-s − 0.0564i·8-s − 0.333·9-s + 0.301·11-s − 0.600i·12-s − 0.651i·13-s + 0.501·14-s − 0.958·16-s − 0.615i·17-s − 0.476i·18-s + 0.0780·19-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

\(L(2)\) \(\approx\) \(1.671974921\)
\(L(\frac12)\) \(\approx\) \(1.671974921\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.03iT - 8T^{2} \)
7 \( 1 + 6.49iT - 343T^{2} \)
13 \( 1 + 30.5iT - 2.19e3T^{2} \)
17 \( 1 + 43.1iT - 4.91e3T^{2} \)
19 \( 1 - 6.46T + 6.85e3T^{2} \)
23 \( 1 + 108. iT - 1.21e4T^{2} \)
29 \( 1 - 274.T + 2.43e4T^{2} \)
31 \( 1 + 68.5T + 2.97e4T^{2} \)
37 \( 1 + 402. iT - 5.06e4T^{2} \)
41 \( 1 + 268.T + 6.89e4T^{2} \)
43 \( 1 - 30.5iT - 7.95e4T^{2} \)
47 \( 1 - 31.5iT - 1.03e5T^{2} \)
53 \( 1 + 252. iT - 1.48e5T^{2} \)
59 \( 1 - 558.T + 2.05e5T^{2} \)
61 \( 1 - 335.T + 2.26e5T^{2} \)
67 \( 1 - 28.7iT - 3.00e5T^{2} \)
71 \( 1 - 89.0T + 3.57e5T^{2} \)
73 \( 1 - 717. iT - 3.89e5T^{2} \)
79 \( 1 + 200.T + 4.93e5T^{2} \)
83 \( 1 + 243. iT - 5.71e5T^{2} \)
89 \( 1 - 312.T + 7.04e5T^{2} \)
97 \( 1 - 27.9iT - 9.12e5T^{2} \)
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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

−9.845037471935401790126314681918, −8.822570472452850101987270472217, −8.274355096570336492722279469190, −7.27417633653381976495122233457, −6.60591829490830876988371092763, −5.62023954396384227867963650085, −4.88798085282220026819920151983, −3.93532243788376720109143808828, −2.56703693922827706083367213764, −0.51473467988473173932800239605, 1.04741049333114436465454450466, 1.89877290027218116958942374658, 2.92344882352631442745012633099, 3.88538186308967808003943169684, 4.98790236595588346133155718159, 6.27717464459518605422831793069, 7.02357200474131996845088783104, 8.277840037737133366769833192639, 9.002858002388987641067336801170, 9.897227882160725086121588474102

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