↑

Properties

Label 2-45-45.29-c4-0-12
Degree $2$
Conductor $45$
Sign $0.999 + 0.0308i$
Analytic cond. $4.65164$
Root an. cond. $2.15676$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.547 − 0.947i)2-s + (6.76 − 5.94i)3-s + (7.40 + 12.8i)4-s + (6.73 + 24.0i)5-s + (−1.93 − 9.65i)6-s + (16.9 + 9.76i)7-s + 33.7·8-s + (10.4 − 80.3i)9-s + (26.5 + 6.79i)10-s + (−100. − 57.9i)11-s + (126. + 42.6i)12-s + (158. − 91.6i)13-s + (18.5 − 10.6i)14-s + (188. + 122. i)15-s + (−99.9 + 173. i)16-s − 89.0·17-s + ⋯
L(s)  = 1  + (0.136 − 0.236i)2-s + (0.751 − 0.660i)3-s + (0.462 + 0.801i)4-s + (0.269 + 0.963i)5-s + (−0.0536 − 0.268i)6-s + (0.345 + 0.199i)7-s + 0.526·8-s + (0.128 − 0.991i)9-s + (0.265 + 0.0679i)10-s + (−0.829 − 0.479i)11-s + (0.876 + 0.296i)12-s + (0.939 − 0.542i)13-s + (0.0944 − 0.0545i)14-s + (0.837 + 0.545i)15-s + (−0.390 + 0.676i)16-s − 0.308·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0308i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 + 0.0308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.999 + 0.0308i$
Analytic conductor: \(4.65164\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :2),\ 0.999 + 0.0308i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.26610 - 0.0350024i\)
\(L(\frac12)\) \(\approx\) \(2.26610 - 0.0350024i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-6.76 + 5.94i)T \)
5 \( 1 + (-6.73 - 24.0i)T \)
good2 \( 1 + (-0.547 + 0.947i)T + (-8 - 13.8i)T^{2} \)
7 \( 1 + (-16.9 - 9.76i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (100. + 57.9i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-158. + 91.6i)T + (1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + 89.0T + 8.35e4T^{2} \)
19 \( 1 + 52.2T + 1.30e5T^{2} \)
23 \( 1 + (232. + 401. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (1.25e3 + 724. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-767. - 1.33e3i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 641. iT - 1.87e6T^{2} \)
41 \( 1 + (1.19e3 - 690. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-1.77e3 - 1.02e3i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-589. + 1.02e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 64.4T + 7.89e6T^{2} \)
59 \( 1 + (-2.83e3 + 1.63e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-929. + 1.60e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (7.25e3 - 4.19e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 4.56e3iT - 2.54e7T^{2} \)
73 \( 1 + 8.07e3iT - 2.83e7T^{2} \)
79 \( 1 + (-4.32e3 + 7.49e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (1.67e3 - 2.89e3i)T + (-2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 - 1.01e4iT - 6.27e7T^{2} \)
97 \( 1 + (-1.18e4 - 6.85e3i)T + (4.42e7 + 7.66e7i)T^{2} \)
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

−14.95820321649002209174965865858, −13.70179118869866424639981425756, −12.94618192130184287297696297743, −11.57995274869380427436470563043, −10.49806065531248756652776659741, −8.538778054110527582505309216730, −7.59852359383925505301712410557, −6.27904757519187141375703297447, −3.45866976365837962967211178632, −2.26675695397519265307761436772, 1.84946257842503522330318812114, 4.39237804904823461477304282215, 5.64306554025185021354460109972, 7.65101916413503970629728289412, 9.031939970432420873742146215536, 10.11817063378420362080343167632, 11.28395107866700122616720299266, 13.18942709602663981291727906320, 13.99245624225006735229304279748, 15.25367097063930311760788658579

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