Properties

Label 2-1800-5.4-c3-0-21
Degree $2$
Conductor $1800$
Sign $0.447 - 0.894i$
Analytic cond. $106.203$
Root an. cond. $10.3055$
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  + 19i·7-s − 22·11-s + i·13-s − 58i·17-s + 53·19-s − 58i·23-s + 22·29-s − 35·31-s + 270i·37-s + 468·41-s − 431i·43-s − 230i·47-s − 18·49-s + 446·59-s + 127·61-s + ⋯
L(s)  = 1  + 1.02i·7-s − 0.603·11-s + 0.0213i·13-s − 0.827i·17-s + 0.639·19-s − 0.525i·23-s + 0.140·29-s − 0.202·31-s + 1.19i·37-s + 1.78·41-s − 1.52i·43-s − 0.713i·47-s − 0.0524·49-s + 0.984·59-s + 0.266·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(106.203\)
Root analytic conductor: \(10.3055\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :3/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.811979851\)
\(L(\frac12)\) \(\approx\) \(1.811979851\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 19iT - 343T^{2} \)
11 \( 1 + 22T + 1.33e3T^{2} \)
13 \( 1 - iT - 2.19e3T^{2} \)
17 \( 1 + 58iT - 4.91e3T^{2} \)
19 \( 1 - 53T + 6.85e3T^{2} \)
23 \( 1 + 58iT - 1.21e4T^{2} \)
29 \( 1 - 22T + 2.43e4T^{2} \)
31 \( 1 + 35T + 2.97e4T^{2} \)
37 \( 1 - 270iT - 5.06e4T^{2} \)
41 \( 1 - 468T + 6.89e4T^{2} \)
43 \( 1 + 431iT - 7.95e4T^{2} \)
47 \( 1 + 230iT - 1.03e5T^{2} \)
53 \( 1 - 1.48e5T^{2} \)
59 \( 1 - 446T + 2.05e5T^{2} \)
61 \( 1 - 127T + 2.26e5T^{2} \)
67 \( 1 - 811iT - 3.00e5T^{2} \)
71 \( 1 + 36T + 3.57e5T^{2} \)
73 \( 1 - 522iT - 3.89e5T^{2} \)
79 \( 1 + 1.36e3T + 4.93e5T^{2} \)
83 \( 1 - 1.13e3iT - 5.71e5T^{2} \)
89 \( 1 - 144T + 7.04e5T^{2} \)
97 \( 1 - 1.07e3iT - 9.12e5T^{2} \)
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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

−8.978953864563239822186012059711, −8.386410561804469857769750618171, −7.47625665065262454068042498570, −6.71394656566363103196436034771, −5.62838586415054428676283680000, −5.21112062651992409561914800821, −4.10577686294779830313008645946, −2.88938623916626015743178619688, −2.28651331479010322288551122849, −0.840988129822077940071024306653, 0.49358286890047522587309732690, 1.56349116405121224044442575562, 2.82882624961970486927670791969, 3.80897897010173835143947235725, 4.56214095566332762806990356334, 5.59283556204901695296822275329, 6.34742313617586669267331525610, 7.54008219943278856057609697814, 7.64580490535863809511218551226, 8.797085239517514994844994507050

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