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Properties

Label 2-70-35.18-c2-0-3
Degree 22
Conductor 7070
Sign 0.969+0.245i0.969 + 0.245i
Analytic cond. 1.907361.90736
Root an. cond. 1.381071.38107
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 0.366i)2-s + (4.90 − 1.31i)3-s + (1.73 + i)4-s + (2.86 + 4.09i)5-s − 7.18·6-s + (−6.99 + 0.145i)7-s + (−1.99 − 2i)8-s + (14.5 − 8.41i)9-s + (−2.41 − 6.64i)10-s + (0.474 − 0.821i)11-s + (9.81 + 2.63i)12-s + (0.862 + 0.862i)13-s + (9.61 + 2.36i)14-s + (19.4 + 16.3i)15-s + (1.99 + 3.46i)16-s + (−7.87 − 29.3i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (1.63 − 0.438i)3-s + (0.433 + 0.250i)4-s + (0.572 + 0.819i)5-s − 1.19·6-s + (−0.999 + 0.0207i)7-s + (−0.249 − 0.250i)8-s + (1.61 − 0.934i)9-s + (−0.241 − 0.664i)10-s + (0.0431 − 0.0746i)11-s + (0.818 + 0.219i)12-s + (0.0663 + 0.0663i)13-s + (0.686 + 0.168i)14-s + (1.29 + 1.09i)15-s + (0.124 + 0.216i)16-s + (−0.463 − 1.72i)17-s + ⋯

Functional equation

Λ(s)=(70s/2 ΓC(s) L(s)=((0.969+0.245i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(70s/2 ΓC(s+1) L(s)=((0.969+0.245i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7070    =    2⋅5⋅72 \cdot 5 \cdot 7
Sign: 0.969+0.245i0.969 + 0.245i
Analytic conductor: 1.907361.90736
Root analytic conductor: 1.381071.38107
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ70(53,⋅)\chi_{70} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 70, ( :1), 0.969+0.245i)(2,\ 70,\ (\ :1),\ 0.969 + 0.245i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.42231−0.177332i1.42231 - 0.177332i
L(12)L(\frac12) ≈\approx 1.42231−0.177332i1.42231 - 0.177332i
L(2)L(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+(1.36+0.366i)T 1 + (1.36 + 0.366i)T
5 1+(−2.86−4.09i)T 1 + (-2.86 - 4.09i)T
7 1+(6.99−0.145i)T 1 + (6.99 - 0.145i)T
good3 1+(−4.90+1.31i)T+(7.79−4.5i)T2 1 + (-4.90 + 1.31i)T + (7.79 - 4.5i)T^{2}
11 1+(−0.474+0.821i)T+(−60.5−104.i)T2 1 + (-0.474 + 0.821i)T + (-60.5 - 104. i)T^{2}
13 1+(−0.862−0.862i)T+169iT2 1 + (-0.862 - 0.862i)T + 169iT^{2}
17 1+(7.87+29.3i)T+(−250.+144.5i)T2 1 + (7.87 + 29.3i)T + (-250. + 144.5i)T^{2}
19 1+(20.9−12.0i)T+(180.5−312.i)T2 1 + (20.9 - 12.0i)T + (180.5 - 312. i)T^{2}
23 1+(−1.46+5.47i)T+(−458.−264.5i)T2 1 + (-1.46 + 5.47i)T + (-458. - 264.5i)T^{2}
29 1−7.33iT−841T2 1 - 7.33iT - 841T^{2}
31 1+(23.5−40.7i)T+(−480.5−832.i)T2 1 + (23.5 - 40.7i)T + (-480.5 - 832. i)T^{2}
37 1+(−7.95−2.13i)T+(1.18e3+684.5i)T2 1 + (-7.95 - 2.13i)T + (1.18e3 + 684.5i)T^{2}
41 1−53.3T+1.68e3T2 1 - 53.3T + 1.68e3T^{2}
43 1+(33.0+33.0i)T+1.84e3iT2 1 + (33.0 + 33.0i)T + 1.84e3iT^{2}
47 1+(−29.3−7.87i)T+(1.91e3+1.10e3i)T2 1 + (-29.3 - 7.87i)T + (1.91e3 + 1.10e3i)T^{2}
53 1+(12.9−3.48i)T+(2.43e3−1.40e3i)T2 1 + (12.9 - 3.48i)T + (2.43e3 - 1.40e3i)T^{2}
59 1+(43.5+25.1i)T+(1.74e3+3.01e3i)T2 1 + (43.5 + 25.1i)T + (1.74e3 + 3.01e3i)T^{2}
61 1+(−22.7−39.4i)T+(−1.86e3+3.22e3i)T2 1 + (-22.7 - 39.4i)T + (-1.86e3 + 3.22e3i)T^{2}
67 1+(17.6+65.9i)T+(−3.88e3+2.24e3i)T2 1 + (17.6 + 65.9i)T + (-3.88e3 + 2.24e3i)T^{2}
71 1+11.9T+5.04e3T2 1 + 11.9T + 5.04e3T^{2}
73 1+(−53.2+14.2i)T+(4.61e3−2.66e3i)T2 1 + (-53.2 + 14.2i)T + (4.61e3 - 2.66e3i)T^{2}
79 1+(−61.4+35.4i)T+(3.12e3−5.40e3i)T2 1 + (-61.4 + 35.4i)T + (3.12e3 - 5.40e3i)T^{2}
83 1+(−85.7−85.7i)T+6.88e3iT2 1 + (-85.7 - 85.7i)T + 6.88e3iT^{2}
89 1+(−135.+78.3i)T+(3.96e3−6.85e3i)T2 1 + (-135. + 78.3i)T + (3.96e3 - 6.85e3i)T^{2}
97 1+(87.1−87.1i)T−9.40e3iT2 1 + (87.1 - 87.1i)T - 9.40e3iT^{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

−14.29776659717467816845530974326, −13.50916917351319309319134881815, −12.45407798790169573956768653380, −10.67594420002303701424314263184, −9.546914059772159157398455348711, −8.884404288329787593825309535983, −7.43025617834802589623578430434, −6.56047038840215223481132079219, −3.35084009219014865229468866165, −2.30107556862150294027785184457, 2.23058043713398732171991754249, 4.04290983837529895463820598660, 6.24515346848631143152287669632, 7.951564780155934398986612779487, 8.920201355310951771211896086499, 9.523159567859842214171236574170, 10.56543778938090917423711253794, 12.80849797415061814651163357539, 13.35575583362676358170283050119, 14.74835414293624582101730700213

Graph of the ZZ-function along the critical line