↑

Properties

Label 2-45-15.14-c8-0-4
Degree $2$
Conductor $45$
Sign $-0.496 - 0.867i$
Analytic cond. $18.3320$
Root an. cond. $4.28159$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 22.2·2-s + 240.·4-s + (−622. − 59.7i)5-s + 2.64e3i·7-s − 350.·8-s + (−1.38e4 − 1.33e3i)10-s + 2.24e4i·11-s + 1.10e4i·13-s + 5.89e4i·14-s − 6.93e4·16-s + 3.55e4·17-s + 3.03e3·19-s + (−1.49e5 − 1.43e4i)20-s + 5.00e5i·22-s − 3.52e5·23-s + ⋯
L(s)  = 1  + 1.39·2-s + 0.938·4-s + (−0.995 − 0.0956i)5-s + 1.10i·7-s − 0.0854·8-s + (−1.38 − 0.133i)10-s + 1.53i·11-s + 0.387i·13-s + 1.53i·14-s − 1.05·16-s + 0.425·17-s + 0.0232·19-s + (−0.934 − 0.0898i)20-s + 2.13i·22-s − 1.26·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 - 0.867i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.496 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.496 - 0.867i$
Analytic conductor: \(18.3320\)
Root analytic conductor: \(4.28159\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :4),\ -0.496 - 0.867i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.04773 + 1.80649i\)
\(L(\frac12)\) \(\approx\) \(1.04773 + 1.80649i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (622. + 59.7i)T \)
good2 \( 1 - 22.2T + 256T^{2} \)
7 \( 1 - 2.64e3iT - 5.76e6T^{2} \)
11 \( 1 - 2.24e4iT - 2.14e8T^{2} \)
13 \( 1 - 1.10e4iT - 8.15e8T^{2} \)
17 \( 1 - 3.55e4T + 6.97e9T^{2} \)
19 \( 1 - 3.03e3T + 1.69e10T^{2} \)
23 \( 1 + 3.52e5T + 7.83e10T^{2} \)
29 \( 1 + 1.14e6iT - 5.00e11T^{2} \)
31 \( 1 - 3.17e5T + 8.52e11T^{2} \)
37 \( 1 - 1.94e6iT - 3.51e12T^{2} \)
41 \( 1 - 3.08e6iT - 7.98e12T^{2} \)
43 \( 1 - 2.55e5iT - 1.16e13T^{2} \)
47 \( 1 + 5.90e6T + 2.38e13T^{2} \)
53 \( 1 - 9.19e6T + 6.22e13T^{2} \)
59 \( 1 + 1.32e7iT - 1.46e14T^{2} \)
61 \( 1 - 2.48e7T + 1.91e14T^{2} \)
67 \( 1 - 3.54e7iT - 4.06e14T^{2} \)
71 \( 1 + 4.67e7iT - 6.45e14T^{2} \)
73 \( 1 + 4.36e6iT - 8.06e14T^{2} \)
79 \( 1 - 1.73e7T + 1.51e15T^{2} \)
83 \( 1 - 5.54e6T + 2.25e15T^{2} \)
89 \( 1 - 9.75e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.26e8iT - 7.83e15T^{2} \)
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

−14.68177215398944543330974731649, −13.19190861800612377000900216309, −12.03077984204975159395307255033, −11.80723085675371872379549475435, −9.673761987681360693273060773525, −8.075195311517223635582034716597, −6.49981890994560616235751011974, −5.02463622329861543393210947143, −3.98088737804097238895178151688, −2.37485070323099338802702135929, 0.48082746447867660408786845267, 3.27233579987509597279805871981, 4.05202987776605661449413829533, 5.58595508783701352705329886512, 7.06974639768160553288876251670, 8.454868310143943634116552646188, 10.59462645237675731612725348286, 11.57864653336157139148421115361, 12.69434731515444185019004176351, 13.82643421574079578655318251768

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