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Properties

Label 2-70-5.4-c5-0-9
Degree 22
Conductor 7070
Sign 0.776+0.630i0.776 + 0.630i
Analytic cond. 11.226811.2268
Root an. cond. 3.350653.35065
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  + 4i·2-s − 22.6i·3-s − 16·4-s + (43.3 + 35.2i)5-s + 90.7·6-s + 49i·7-s − 64i·8-s − 272.·9-s + (−141. + 173. i)10-s + 527.·11-s + 363. i·12-s − 1.08e3i·13-s − 196·14-s + (800. − 984. i)15-s + 256·16-s − 1.82e3i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.45i·3-s − 0.5·4-s + (0.776 + 0.630i)5-s + 1.02·6-s + 0.377i·7-s − 0.353i·8-s − 1.11·9-s + (−0.445 + 0.548i)10-s + 1.31·11-s + 0.727i·12-s − 1.77i·13-s − 0.267·14-s + (0.918 − 1.12i)15-s + 0.250·16-s − 1.53i·17-s + ⋯

Functional equation

Λ(s)=(70s/2 ΓC(s) L(s)=((0.776+0.630i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(70s/2 ΓC(s+5/2) L(s)=((0.776+0.630i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7070    =    2⋅5⋅72 \cdot 5 \cdot 7
Sign: 0.776+0.630i0.776 + 0.630i
Analytic conductor: 11.226811.2268
Root analytic conductor: 3.350653.35065
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ70(29,⋅)\chi_{70} (29, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 70, ( :5/2), 0.776+0.630i)(2,\ 70,\ (\ :5/2),\ 0.776 + 0.630i)

Particular Values

L(3)L(3) ≈\approx 1.80762−0.641812i1.80762 - 0.641812i
L(12)L(\frac12) ≈\approx 1.80762−0.641812i1.80762 - 0.641812i
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 1−4iT 1 - 4iT
5 1+(−43.3−35.2i)T 1 + (-43.3 - 35.2i)T
7 1−49iT 1 - 49iT
good3 1+22.6iT−243T2 1 + 22.6iT - 243T^{2}
11 1−527.T+1.61e5T2 1 - 527.T + 1.61e5T^{2}
13 1+1.08e3iT−3.71e5T2 1 + 1.08e3iT - 3.71e5T^{2}
17 1+1.82e3iT−1.41e6T2 1 + 1.82e3iT - 1.41e6T^{2}
19 1−113.T+2.47e6T2 1 - 113.T + 2.47e6T^{2}
23 1−8.45iT−6.43e6T2 1 - 8.45iT - 6.43e6T^{2}
29 1−7.75e3T+2.05e7T2 1 - 7.75e3T + 2.05e7T^{2}
31 1+5.07e3T+2.86e7T2 1 + 5.07e3T + 2.86e7T^{2}
37 1−1.12e4iT−6.93e7T2 1 - 1.12e4iT - 6.93e7T^{2}
41 1−4.49e3T+1.15e8T2 1 - 4.49e3T + 1.15e8T^{2}
43 1+1.71e4iT−1.47e8T2 1 + 1.71e4iT - 1.47e8T^{2}
47 1−1.65e3iT−2.29e8T2 1 - 1.65e3iT - 2.29e8T^{2}
53 1−6.20e3iT−4.18e8T2 1 - 6.20e3iT - 4.18e8T^{2}
59 1+6.32e3T+7.14e8T2 1 + 6.32e3T + 7.14e8T^{2}
61 1−2.28e4T+8.44e8T2 1 - 2.28e4T + 8.44e8T^{2}
67 1−6.60e4iT−1.35e9T2 1 - 6.60e4iT - 1.35e9T^{2}
71 1+2.98e4T+1.80e9T2 1 + 2.98e4T + 1.80e9T^{2}
73 1−2.01e4iT−2.07e9T2 1 - 2.01e4iT - 2.07e9T^{2}
79 1+2.27e4T+3.07e9T2 1 + 2.27e4T + 3.07e9T^{2}
83 1−8.15e4iT−3.93e9T2 1 - 8.15e4iT - 3.93e9T^{2}
89 1+7.01e4T+5.58e9T2 1 + 7.01e4T + 5.58e9T^{2}
97 1+9.37e4iT−8.58e9T2 1 + 9.37e4iT - 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

−13.72456588792517806966711737774, −12.74679281313366293520625524818, −11.69313174057816816675008232791, −10.01293081228602045489840377939, −8.662666024089320744573184540762, −7.35497334903504880372293437000, −6.55084057257130076573251874012, −5.49575929342288443442393764266, −2.80566782478734642695123587448, −0.992084140478228599584514691318, 1.61149857869042726655174136103, 3.88243140363068982158760594627, 4.60372037360518008425324630800, 6.27020317734126453716070164450, 8.809272923373119837507441143511, 9.390976237854387166037473728439, 10.33452324337145829531893188336, 11.39031198771377445161699193733, 12.59817541572690969187889117463, 14.04578028264643251466809573307

Graph of the ZZ-function along the critical line