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Properties

Label 2-70-35.13-c1-0-1
Degree 22
Conductor 7070
Sign 0.979+0.201i0.979 + 0.201i
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  + (−0.707 + 0.707i)2-s + (1.30 − 1.30i)3-s − 1.00i·4-s + (0.158 − 2.23i)5-s + 1.84i·6-s + (−0.941 + 2.47i)7-s + (0.707 + 0.707i)8-s − 0.414i·9-s + (1.46 + 1.68i)10-s + 2.82·11-s + (−1.30 − 1.30i)12-s + (−4.23 + 4.23i)13-s + (−1.08 − 2.41i)14-s + (−2.70 − 3.12i)15-s − 1.00·16-s + (−3.69 − 3.69i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.754 − 0.754i)3-s − 0.500i·4-s + (0.0708 − 0.997i)5-s + 0.754i·6-s + (−0.355 + 0.934i)7-s + (0.250 + 0.250i)8-s − 0.138i·9-s + (0.463 + 0.534i)10-s + 0.852·11-s + (−0.377 − 0.377i)12-s + (−1.17 + 1.17i)13-s + (−0.289 − 0.645i)14-s + (−0.698 − 0.805i)15-s − 0.250·16-s + (−0.896 − 0.896i)17-s + ⋯

Functional equation

Λ(s)=(70s/2 ΓC(s) L(s)=((0.979+0.201i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.201i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(70s/2 ΓC(s+1/2) L(s)=((0.979+0.201i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7070    =    2⋅5⋅72 \cdot 5 \cdot 7
Sign: 0.979+0.201i0.979 + 0.201i
Analytic conductor: 0.5589520.558952
Root analytic conductor: 0.7476310.747631
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ70(13,⋅)\chi_{70} (13, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 70, ( :1/2), 0.979+0.201i)(2,\ 70,\ (\ :1/2),\ 0.979 + 0.201i)

Particular Values

L(1)L(1) ≈\approx 0.866432−0.0882676i0.866432 - 0.0882676i
L(12)L(\frac12) ≈\approx 0.866432−0.0882676i0.866432 - 0.0882676i
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+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
5 1+(−0.158+2.23i)T 1 + (-0.158 + 2.23i)T
7 1+(0.941−2.47i)T 1 + (0.941 - 2.47i)T
good3 1+(−1.30+1.30i)T−3iT2 1 + (-1.30 + 1.30i)T - 3iT^{2}
11 1−2.82T+11T2 1 - 2.82T + 11T^{2}
13 1+(4.23−4.23i)T−13iT2 1 + (4.23 - 4.23i)T - 13iT^{2}
17 1+(3.69+3.69i)T+17iT2 1 + (3.69 + 3.69i)T + 17iT^{2}
19 1−1.39T+19T2 1 - 1.39T + 19T^{2}
23 1+(−0.414−0.414i)T+23iT2 1 + (-0.414 - 0.414i)T + 23iT^{2}
29 1−0.828iT−29T2 1 - 0.828iT - 29T^{2}
31 1+1.53iT−31T2 1 + 1.53iT - 31T^{2}
37 1+(−2.58+2.58i)T−37iT2 1 + (-2.58 + 2.58i)T - 37iT^{2}
41 1+3.69iT−41T2 1 + 3.69iT - 41T^{2}
43 1+(−4−4i)T+43iT2 1 + (-4 - 4i)T + 43iT^{2}
47 1+(−1.08−1.08i)T+47iT2 1 + (-1.08 - 1.08i)T + 47iT^{2}
53 1+(8.24+8.24i)T+53iT2 1 + (8.24 + 8.24i)T + 53iT^{2}
59 1+9.23T+59T2 1 + 9.23T + 59T^{2}
61 1−6.43iT−61T2 1 - 6.43iT - 61T^{2}
67 1+(−10.4+10.4i)T−67iT2 1 + (-10.4 + 10.4i)T - 67iT^{2}
71 1+0.585T+71T2 1 + 0.585T + 71T^{2}
73 1+(−4.14+4.14i)T−73iT2 1 + (-4.14 + 4.14i)T - 73iT^{2}
79 1+5.07iT−79T2 1 + 5.07iT - 79T^{2}
83 1+(−5.31+5.31i)T−83iT2 1 + (-5.31 + 5.31i)T - 83iT^{2}
89 1−11.3T+89T2 1 - 11.3T + 89T^{2}
97 1+(4.59+4.59i)T+97iT2 1 + (4.59 + 4.59i)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

−14.59657449981060710725166800223, −13.73460963481235310312476185764, −12.58552378802120746593658974618, −11.63020594915862570810532593404, −9.303791439939596796170436166651, −9.124475990955019392277946217372, −7.75192276755752972122424539104, −6.56757197148260786786250918424, −4.88199365542020945463572457055, −2.11189885120549725243992584967, 2.93893812303338789770631387506, 4.07963943454371272144407811349, 6.65806377191328104629730671593, 7.921038775700212950159562894875, 9.409940639487214685885581009183, 10.14868686409161879514923571805, 10.99134125831819050218991468451, 12.50552335021490797127893041958, 13.85977008777389613366498635225, 14.78470987378120487362941040263

Graph of the ZZ-function along the critical line