↑

Properties

Label 2-90-5.3-c6-0-2
Degree 22
Conductor 9090
Sign −0.767−0.640i-0.767 - 0.640i
Analytic cond. 20.704820.7048
Root an. cond. 4.550264.55026
Motivic weight 66
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4 − 4i)2-s − 32i·4-s + (75 + 100i)5-s + (−247 + 247i)7-s + (−128 − 128i)8-s + (700 + 100i)10-s − 1.40e3·11-s + (−2.70e3 − 2.70e3i)13-s + 1.97e3i·14-s − 1.02e3·16-s + (−2.59e3 + 2.59e3i)17-s − 1.72e3i·19-s + (3.20e3 − 2.40e3i)20-s + (−5.60e3 + 5.60e3i)22-s + (−2.13e3 − 2.13e3i)23-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s − 0.5i·4-s + (0.599 + 0.800i)5-s + (−0.720 + 0.720i)7-s + (−0.250 − 0.250i)8-s + (0.700 + 0.100i)10-s − 1.05·11-s + (−1.23 − 1.23i)13-s + 0.720i·14-s − 0.250·16-s + (−0.527 + 0.527i)17-s − 0.250i·19-s + (0.400 − 0.299i)20-s + (−0.526 + 0.526i)22-s + (−0.175 − 0.175i)23-s + ⋯

Functional equation

Λ(s)=(90s/2 ΓC(s) L(s)=((−0.767−0.640i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 - 0.640i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(90s/2 ΓC(s+3) L(s)=((−0.767−0.640i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9090    =    2⋅32⋅52 \cdot 3^{2} \cdot 5
Sign: −0.767−0.640i-0.767 - 0.640i
Analytic conductor: 20.704820.7048
Root analytic conductor: 4.550264.55026
Motivic weight: 66
Rational: no
Arithmetic: yes
Character: χ90(73,⋅)\chi_{90} (73, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 90, ( :3), −0.767−0.640i)(2,\ 90,\ (\ :3),\ -0.767 - 0.640i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 0.146057+0.402957i0.146057 + 0.402957i
L(12)L(\frac12) ≈\approx 0.146057+0.402957i0.146057 + 0.402957i
L(4)L(4) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−4+4i)T 1 + (-4 + 4i)T
3 1 1
5 1+(−75−100i)T 1 + (-75 - 100i)T
good7 1+(247−247i)T−1.17e5iT2 1 + (247 - 247i)T - 1.17e5iT^{2}
11 1+1.40e3T+1.77e6T2 1 + 1.40e3T + 1.77e6T^{2}
13 1+(2.70e3+2.70e3i)T+4.82e6iT2 1 + (2.70e3 + 2.70e3i)T + 4.82e6iT^{2}
17 1+(2.59e3−2.59e3i)T−2.41e7iT2 1 + (2.59e3 - 2.59e3i)T - 2.41e7iT^{2}
19 1+1.72e3iT−4.70e7T2 1 + 1.72e3iT - 4.70e7T^{2}
23 1+(2.13e3+2.13e3i)T+1.48e8iT2 1 + (2.13e3 + 2.13e3i)T + 1.48e8iT^{2}
29 1−3.05e4iT−5.94e8T2 1 - 3.05e4iT - 5.94e8T^{2}
31 1+3.78e4T+8.87e8T2 1 + 3.78e4T + 8.87e8T^{2}
37 1+(−3.71e4+3.71e4i)T−2.56e9iT2 1 + (-3.71e4 + 3.71e4i)T - 2.56e9iT^{2}
41 1−3.54e4T+4.75e9T2 1 - 3.54e4T + 4.75e9T^{2}
43 1+(−3.91e4−3.91e4i)T+6.32e9iT2 1 + (-3.91e4 - 3.91e4i)T + 6.32e9iT^{2}
47 1+(9.51e4−9.51e4i)T−1.07e10iT2 1 + (9.51e4 - 9.51e4i)T - 1.07e10iT^{2}
53 1+(3.60e4+3.60e4i)T+2.21e10iT2 1 + (3.60e4 + 3.60e4i)T + 2.21e10iT^{2}
59 1+3.59e4iT−4.21e10T2 1 + 3.59e4iT - 4.21e10T^{2}
61 1−8.33e4T+5.15e10T2 1 - 8.33e4T + 5.15e10T^{2}
67 1+(−6.08e4+6.08e4i)T−9.04e10iT2 1 + (-6.08e4 + 6.08e4i)T - 9.04e10iT^{2}
71 1−4.03e4T+1.28e11T2 1 - 4.03e4T + 1.28e11T^{2}
73 1+(1.29e5+1.29e5i)T+1.51e11iT2 1 + (1.29e5 + 1.29e5i)T + 1.51e11iT^{2}
79 1+5.24e5iT−2.43e11T2 1 + 5.24e5iT - 2.43e11T^{2}
83 1+(−1.14e5−1.14e5i)T+3.26e11iT2 1 + (-1.14e5 - 1.14e5i)T + 3.26e11iT^{2}
89 1−1.87e5iT−4.96e11T2 1 - 1.87e5iT - 4.96e11T^{2}
97 1+(−5.32e5+5.32e5i)T−8.32e11iT2 1 + (-5.32e5 + 5.32e5i)T - 8.32e11iT^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−12.90659162306233259521310846468, −12.72080597963365331754577819678, −11.04082163577976276153011677282, −10.26051239797065738713836479426, −9.284889288292758594728136392404, −7.52732887570071554203270932564, −6.11846653768333266733256589490, −5.16219390518300436757912977727, −3.12002662920330230389779545959, −2.31554495773752701089792040816, 0.11476818580777645209833985475, 2.32740584311359197384466324035, 4.21443480897899988040819078917, 5.29347774804405597401155189830, 6.64327330575763261087224963082, 7.73875061616226159097258905986, 9.231953706310146854747662293979, 10.08364927269218266355360909840, 11.70760239852399642527844274708, 12.87212511669701196809198595644

Graph of the ZZ-function along the critical line