↑

Properties

Label 2-45-45.34-c3-0-4
Degree $2$
Conductor $45$
Sign $0.100 + 0.994i$
Analytic cond. $2.65508$
Root an. cond. $1.62944$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−4.35 − 2.51i)2-s + (−1.99 + 4.79i)3-s + (8.61 + 14.9i)4-s + (5.42 − 9.77i)5-s + (20.7 − 15.8i)6-s + (−4.66 − 2.69i)7-s − 46.4i·8-s + (−19.0 − 19.1i)9-s + (−48.1 + 28.9i)10-s + (19.5 − 33.8i)11-s + (−88.8 + 11.5i)12-s + (75.0 − 43.3i)13-s + (13.5 + 23.4i)14-s + (36.0 + 45.5i)15-s + (−47.6 + 82.4i)16-s − 15.4i·17-s + ⋯
L(s)  = 1  + (−1.53 − 0.888i)2-s + (−0.384 + 0.923i)3-s + (1.07 + 1.86i)4-s + (0.485 − 0.874i)5-s + (1.41 − 1.07i)6-s + (−0.251 − 0.145i)7-s − 2.05i·8-s + (−0.704 − 0.709i)9-s + (−1.52 + 0.914i)10-s + (0.535 − 0.928i)11-s + (−2.13 + 0.277i)12-s + (1.60 − 0.924i)13-s + (0.258 + 0.446i)14-s + (0.621 + 0.783i)15-s + (−0.744 + 1.28i)16-s − 0.219i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.100 + 0.994i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ 0.100 + 0.994i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.428616 - 0.387316i\)
\(L(\frac12)\) \(\approx\) \(0.428616 - 0.387316i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.99 - 4.79i)T \)
5 \( 1 + (-5.42 + 9.77i)T \)
good2 \( 1 + (4.35 + 2.51i)T + (4 + 6.92i)T^{2} \)
7 \( 1 + (4.66 + 2.69i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-19.5 + 33.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-75.0 + 43.3i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 15.4iT - 4.91e3T^{2} \)
19 \( 1 - 26.8T + 6.85e3T^{2} \)
23 \( 1 + (96.2 - 55.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-24.6 + 42.6i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-89.9 - 155. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 293. iT - 5.06e4T^{2} \)
41 \( 1 + (13.8 + 23.9i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (52.3 + 30.2i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-83.3 - 48.1i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 251. iT - 1.48e5T^{2} \)
59 \( 1 + (38.4 + 66.5i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (245. - 424. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-206. + 119. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 640.T + 3.57e5T^{2} \)
73 \( 1 - 769. iT - 3.89e5T^{2} \)
79 \( 1 + (-331. + 573. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-1.12e3 - 651. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 995.T + 7.04e5T^{2} \)
97 \( 1 + (705. + 407. i)T + (4.56e5 + 7.90e5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.93626518380477734889176205808, −13.67010528379450484748977047365, −12.16648377077050210722719765605, −11.09600460181279294430916527989, −10.16871506475982809649959375009, −9.107107453435322554154989377035, −8.346324738810056823423568399561, −5.90423752918455022133261147525, −3.53125803000424846350781480401, −0.857323424640429621175462167289, 1.68764778875377152742437479782, 6.20747361575528336771140709453, 6.66364052929789985651618665126, 7.991978303478303894561551506238, 9.330370261792162762223741738207, 10.55415218898145411546494099439, 11.70431906384752925905092703535, 13.57529518079421958356178459753, 14.70250893013245055843290998780, 15.97845657179459855813965094866

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