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Properties

Label 2-70-35.12-c1-0-2
Degree 22
Conductor 7070
Sign 0.509+0.860i0.509 + 0.860i
Analytic cond. 0.5589520.558952
Root an. cond. 0.7476310.747631
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.523 − 1.95i)3-s + (0.866 − 0.499i)4-s + (−2.03 − 0.935i)5-s + 2.02i·6-s + (1.83 − 1.90i)7-s + (−0.707 + 0.707i)8-s + (−0.941 − 0.543i)9-s + (2.20 + 0.378i)10-s + (2.01 + 3.49i)11-s + (−0.523 − 1.95i)12-s + (0.204 + 0.204i)13-s + (−1.28 + 2.31i)14-s + (−2.89 + 3.47i)15-s + (0.500 − 0.866i)16-s + (−1.97 − 0.527i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.302 − 1.12i)3-s + (0.433 − 0.249i)4-s + (−0.908 − 0.418i)5-s + 0.825i·6-s + (0.695 − 0.718i)7-s + (−0.249 + 0.249i)8-s + (−0.313 − 0.181i)9-s + (0.696 + 0.119i)10-s + (0.609 + 1.05i)11-s + (−0.151 − 0.563i)12-s + (0.0568 + 0.0568i)13-s + (−0.343 + 0.618i)14-s + (−0.746 + 0.897i)15-s + (0.125 − 0.216i)16-s + (−0.477 − 0.128i)17-s + ⋯

Functional equation

Λ(s)=(70s/2 ΓC(s) L(s)=((0.509+0.860i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(70s/2 ΓC(s+1/2) L(s)=((0.509+0.860i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7070    =    2⋅5⋅72 \cdot 5 \cdot 7
Sign: 0.509+0.860i0.509 + 0.860i
Analytic conductor: 0.5589520.558952
Root analytic conductor: 0.7476310.747631
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ70(47,⋅)\chi_{70} (47, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 70, ( :1/2), 0.509+0.860i)(2,\ 70,\ (\ :1/2),\ 0.509 + 0.860i)

Particular Values

L(1)L(1) ≈\approx 0.622454−0.354602i0.622454 - 0.354602i
L(12)L(\frac12) ≈\approx 0.622454−0.354602i0.622454 - 0.354602i
L(32)L(\frac{3}{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+(0.965−0.258i)T 1 + (0.965 - 0.258i)T
5 1+(2.03+0.935i)T 1 + (2.03 + 0.935i)T
7 1+(−1.83+1.90i)T 1 + (-1.83 + 1.90i)T
good3 1+(−0.523+1.95i)T+(−2.59−1.5i)T2 1 + (-0.523 + 1.95i)T + (-2.59 - 1.5i)T^{2}
11 1+(−2.01−3.49i)T+(−5.5+9.52i)T2 1 + (-2.01 - 3.49i)T + (-5.5 + 9.52i)T^{2}
13 1+(−0.204−0.204i)T+13iT2 1 + (-0.204 - 0.204i)T + 13iT^{2}
17 1+(1.97+0.527i)T+(14.7+8.5i)T2 1 + (1.97 + 0.527i)T + (14.7 + 8.5i)T^{2}
19 1+(3.10−5.37i)T+(−9.5−16.4i)T2 1 + (3.10 - 5.37i)T + (-9.5 - 16.4i)T^{2}
23 1+(−1.17−4.38i)T+(−19.9+11.5i)T2 1 + (-1.17 - 4.38i)T + (-19.9 + 11.5i)T^{2}
29 1+7.15iT−29T2 1 + 7.15iT - 29T^{2}
31 1+(−6.33+3.65i)T+(15.5−26.8i)T2 1 + (-6.33 + 3.65i)T + (15.5 - 26.8i)T^{2}
37 1+(4.46−1.19i)T+(32.0−18.5i)T2 1 + (4.46 - 1.19i)T + (32.0 - 18.5i)T^{2}
41 1−2.58iT−41T2 1 - 2.58iT - 41T^{2}
43 1+(4.97−4.97i)T−43iT2 1 + (4.97 - 4.97i)T - 43iT^{2}
47 1+(0.0815+0.304i)T+(−40.7+23.5i)T2 1 + (0.0815 + 0.304i)T + (-40.7 + 23.5i)T^{2}
53 1+(8.00+2.14i)T+(45.8+26.5i)T2 1 + (8.00 + 2.14i)T + (45.8 + 26.5i)T^{2}
59 1+(−0.427−0.740i)T+(−29.5+51.0i)T2 1 + (-0.427 - 0.740i)T + (-29.5 + 51.0i)T^{2}
61 1+(5.99+3.46i)T+(30.5+52.8i)T2 1 + (5.99 + 3.46i)T + (30.5 + 52.8i)T^{2}
67 1+(0.817−3.05i)T+(−58.0−33.5i)T2 1 + (0.817 - 3.05i)T + (-58.0 - 33.5i)T^{2}
71 1−7.12T+71T2 1 - 7.12T + 71T^{2}
73 1+(2.98−11.1i)T+(−63.2−36.5i)T2 1 + (2.98 - 11.1i)T + (-63.2 - 36.5i)T^{2}
79 1+(4.39+2.53i)T+(39.5+68.4i)T2 1 + (4.39 + 2.53i)T + (39.5 + 68.4i)T^{2}
83 1+(3.85+3.85i)T+83iT2 1 + (3.85 + 3.85i)T + 83iT^{2}
89 1+(1.53−2.66i)T+(−44.5−77.0i)T2 1 + (1.53 - 2.66i)T + (-44.5 - 77.0i)T^{2}
97 1+(6.63−6.63i)T−97iT2 1 + (6.63 - 6.63i)T - 97iT^{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

−14.60890409366785953523866453667, −13.39635362248342307623637135421, −12.24209159715530163845724630915, −11.38988144769205107773740841200, −9.878522828395303252848506428670, −8.286034516612125050687061433734, −7.68324315817142882728372994284, −6.67808905485602647026870735832, −4.36288301637604859248062731712, −1.57751405592454151361540136058, 3.13904817475299052291053449971, 4.63057072172908045307350318820, 6.73844098056840480408485539956, 8.522977572268320191167320686536, 8.908677182057046311054470353093, 10.60151321426083269417982551579, 11.16181088725231799729047903862, 12.31274380923542050071877369425, 14.24219474580634956196800206242, 15.22918413582414997142984669262

Graph of the ZZ-function along the critical line