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Properties

Label 2-42-7.2-c5-0-3
Degree 22
Conductor 4242
Sign 0.266−0.963i0.266 - 0.963i
Analytic cond. 6.736126.73612
Root an. cond. 2.595402.59540
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 3.46i)2-s + (4.5 + 7.79i)3-s + (−7.99 − 13.8i)4-s + (−43 + 74.4i)5-s + 36·6-s + (24.5 + 127. i)7-s − 63.9·8-s + (−40.5 + 70.1i)9-s + (172 + 297. i)10-s + (−17 − 29.4i)11-s + (72 − 124. i)12-s − 3·13-s + (490 + 169. i)14-s − 774.·15-s + (−128 + 221. i)16-s + (952 + 1.64e3i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.769 + 1.33i)5-s + 0.408·6-s + (0.188 + 0.981i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.543 + 0.942i)10-s + (−0.0423 − 0.0733i)11-s + (0.144 − 0.249i)12-s − 0.00492·13-s + (0.668 + 0.231i)14-s − 0.888·15-s + (−0.125 + 0.216i)16-s + (0.798 + 1.38i)17-s + ⋯

Functional equation

Λ(s)=(42s/2 ΓC(s) L(s)=((0.266−0.963i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(42s/2 ΓC(s+5/2) L(s)=((0.266−0.963i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4242    =    2⋅3⋅72 \cdot 3 \cdot 7
Sign: 0.266−0.963i0.266 - 0.963i
Analytic conductor: 6.736126.73612
Root analytic conductor: 2.595402.59540
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ42(37,⋅)\chi_{42} (37, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 42, ( :5/2), 0.266−0.963i)(2,\ 42,\ (\ :5/2),\ 0.266 - 0.963i)

Particular Values

L(3)L(3) ≈\approx 1.27438+0.969495i1.27438 + 0.969495i
L(12)L(\frac12) ≈\approx 1.27438+0.969495i1.27438 + 0.969495i
L(72)L(\frac{7}{2}) 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+(−2+3.46i)T 1 + (-2 + 3.46i)T
3 1+(−4.5−7.79i)T 1 + (-4.5 - 7.79i)T
7 1+(−24.5−127.i)T 1 + (-24.5 - 127. i)T
good5 1+(43−74.4i)T+(−1.56e3−2.70e3i)T2 1 + (43 - 74.4i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(17+29.4i)T+(−8.05e4+1.39e5i)T2 1 + (17 + 29.4i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1+3T+3.71e5T2 1 + 3T + 3.71e5T^{2}
17 1+(−952−1.64e3i)T+(−7.09e5+1.22e6i)T2 1 + (-952 - 1.64e3i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(−744.5+1.28e3i)T+(−1.23e6−2.14e6i)T2 1 + (-744.5 + 1.28e3i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(−112+193.i)T+(−3.21e6−5.57e6i)T2 1 + (-112 + 193. i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1+6.50e3T+2.05e7T2 1 + 6.50e3T + 2.05e7T^{2}
31 1+(865.5+1.49e3i)T+(−1.43e7+2.47e7i)T2 1 + (865.5 + 1.49e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(−3.81e3+6.61e3i)T+(−3.46e7−6.00e7i)T2 1 + (-3.81e3 + 6.61e3i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1−1.54e4T+1.15e8T2 1 - 1.54e4T + 1.15e8T^{2}
43 1−1.84e4T+1.47e8T2 1 - 1.84e4T + 1.47e8T^{2}
47 1+(9.23e3−1.59e4i)T+(−1.14e8−1.98e8i)T2 1 + (9.23e3 - 1.59e4i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(−9.97e3−1.72e4i)T+(−2.09e8+3.62e8i)T2 1 + (-9.97e3 - 1.72e4i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(−1.59e4−2.75e4i)T+(−3.57e8+6.19e8i)T2 1 + (-1.59e4 - 2.75e4i)T + (-3.57e8 + 6.19e8i)T^{2}
61 1+(−2.88e4+4.99e4i)T+(−4.22e8−7.31e8i)T2 1 + (-2.88e4 + 4.99e4i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(−3.02e4−5.24e4i)T+(−6.75e8+1.16e9i)T2 1 + (-3.02e4 - 5.24e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1+4.48e4T+1.80e9T2 1 + 4.48e4T + 1.80e9T^{2}
73 1+(1.04e4+1.80e4i)T+(−1.03e9+1.79e9i)T2 1 + (1.04e4 + 1.80e4i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(−1.52e4+2.64e4i)T+(−1.53e9−2.66e9i)T2 1 + (-1.52e4 + 2.64e4i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1−1.10e5T+3.93e9T2 1 - 1.10e5T + 3.93e9T^{2}
89 1+(−2.94e4+5.10e4i)T+(−2.79e9−4.83e9i)T2 1 + (-2.94e4 + 5.10e4i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1+1.19e5T+8.58e9T2 1 + 1.19e5T + 8.58e9T^{2}
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   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

−14.92620898687039245536545767616, −14.47216167005077790726340839807, −12.75117476496957972854954856916, −11.44387136750234889762405014090, −10.71752267874604716279027041342, −9.284111729075777432914812719963, −7.71143991980332458817599347908, −5.81909624726626437707333953063, −3.87527031054882447815796470880, −2.60770790110591299546685996876, 0.802615122734954768208848093292, 3.85567495749267176241131341764, 5.25792909075009862551866829005, 7.31543270937134885296979933300, 8.056751827785019443898501584076, 9.464709440523692675298340368460, 11.59235423336549650335378403877, 12.62707534485783479501012057098, 13.59533718097096214751554718749, 14.65670869096059589267035278023

Graph of the ZZ-function along the critical line