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Properties

Label 2-30e2-5.4-c3-0-2
Degree $2$
Conductor $900$
Sign $-0.447 - 0.894i$
Analytic cond. $53.1017$
Root an. cond. $7.28709$
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  + 28i·7-s + 24·11-s − 70i·13-s + 102i·17-s − 20·19-s + 72i·23-s + 306·29-s − 136·31-s + 214i·37-s + 150·41-s − 292i·43-s − 72i·47-s − 441·49-s + 414i·53-s − 744·59-s + ⋯
L(s)  = 1  + 1.51i·7-s + 0.657·11-s − 1.49i·13-s + 1.45i·17-s − 0.241·19-s + 0.652i·23-s + 1.95·29-s − 0.787·31-s + 0.950i·37-s + 0.571·41-s − 1.03i·43-s − 0.223i·47-s − 1.28·49-s + 1.07i·53-s − 1.64·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(53.1017\)
Root analytic conductor: \(7.28709\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :3/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.569874801\)
\(L(\frac12)\) \(\approx\) \(1.569874801\)
\(L(\frac{5}{2})\) 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 \( 1 - 28iT - 343T^{2} \)
11 \( 1 - 24T + 1.33e3T^{2} \)
13 \( 1 + 70iT - 2.19e3T^{2} \)
17 \( 1 - 102iT - 4.91e3T^{2} \)
19 \( 1 + 20T + 6.85e3T^{2} \)
23 \( 1 - 72iT - 1.21e4T^{2} \)
29 \( 1 - 306T + 2.43e4T^{2} \)
31 \( 1 + 136T + 2.97e4T^{2} \)
37 \( 1 - 214iT - 5.06e4T^{2} \)
41 \( 1 - 150T + 6.89e4T^{2} \)
43 \( 1 + 292iT - 7.95e4T^{2} \)
47 \( 1 + 72iT - 1.03e5T^{2} \)
53 \( 1 - 414iT - 1.48e5T^{2} \)
59 \( 1 + 744T + 2.05e5T^{2} \)
61 \( 1 + 418T + 2.26e5T^{2} \)
67 \( 1 + 188iT - 3.00e5T^{2} \)
71 \( 1 + 480T + 3.57e5T^{2} \)
73 \( 1 - 434iT - 3.89e5T^{2} \)
79 \( 1 + 1.35e3T + 4.93e5T^{2} \)
83 \( 1 - 612iT - 5.71e5T^{2} \)
89 \( 1 + 30T + 7.04e5T^{2} \)
97 \( 1 - 286iT - 9.12e5T^{2} \)
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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

−10.05148089847768657105512128132, −8.945914702476504252451526719194, −8.493457322375280236180430281444, −7.60998413529574410715563183673, −6.24851270508023085025879226312, −5.82517966506479251971301862736, −4.80612317799707652576654368766, −3.50670120297702414588907478625, −2.56873688172292688081460508705, −1.32433227819911283374793450289, 0.42336048383779257978793941050, 1.55779943933213546956746539738, 3.01550772510457730470272697926, 4.28760867909897646048428809535, 4.61404261012663206986983874987, 6.23966085883748587886819153210, 6.95472081456936610102471306398, 7.52456196476033473134153436482, 8.750259082552496753044344534384, 9.461458189881491980173682712288

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