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Properties

Label 2-70-7.4-c1-0-1
Degree 22
Conductor 7070
Sign 0.386−0.922i0.386 - 0.922i
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s − 0.999·6-s + (−0.5 − 2.59i)7-s − 0.999·8-s + (1 + 1.73i)9-s + (−0.499 + 0.866i)10-s + (3 − 5.19i)11-s + (−0.499 − 0.866i)12-s − 4·13-s + (2 − 1.73i)14-s − 0.999·15-s + (−0.5 − 0.866i)16-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s − 0.408·6-s + (−0.188 − 0.981i)7-s − 0.353·8-s + (0.333 + 0.577i)9-s + (−0.158 + 0.273i)10-s + (0.904 − 1.56i)11-s + (−0.144 − 0.249i)12-s − 1.10·13-s + (0.534 − 0.462i)14-s − 0.258·15-s + (−0.125 − 0.216i)16-s + ⋯

Functional equation

Λ(s)=(70s/2 ΓC(s) L(s)=((0.386−0.922i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(70s/2 ΓC(s+1/2) L(s)=((0.386−0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7070    =    2⋅5⋅72 \cdot 5 \cdot 7
Sign: 0.386−0.922i0.386 - 0.922i
Analytic conductor: 0.5589520.558952
Root analytic conductor: 0.7476310.747631
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ70(11,⋅)\chi_{70} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 70, ( :1/2), 0.386−0.922i)(2,\ 70,\ (\ :1/2),\ 0.386 - 0.922i)

Particular Values

L(1)L(1) ≈\approx 0.827466+0.550415i0.827466 + 0.550415i
L(12)L(\frac12) ≈\approx 0.827466+0.550415i0.827466 + 0.550415i
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+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
5 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
7 1+(0.5+2.59i)T 1 + (0.5 + 2.59i)T
good3 1+(0.5−0.866i)T+(−1.5−2.59i)T2 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2}
11 1+(−3+5.19i)T+(−5.5−9.52i)T2 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2}
13 1+4T+13T2 1 + 4T + 13T^{2}
17 1+(−8.5−14.7i)T2 1 + (-8.5 - 14.7i)T^{2}
19 1+(1+1.73i)T+(−9.5+16.4i)T2 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2}
23 1+(−1.5−2.59i)T+(−11.5+19.9i)T2 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2}
29 1+3T+29T2 1 + 3T + 29T^{2}
31 1+(4−6.92i)T+(−15.5−26.8i)T2 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2}
37 1+(−2−3.46i)T+(−18.5+32.0i)T2 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2}
41 1−9T+41T2 1 - 9T + 41T^{2}
43 1+7T+43T2 1 + 7T + 43T^{2}
47 1+(−23.5+40.7i)T2 1 + (-23.5 + 40.7i)T^{2}
53 1+(−3+5.19i)T+(−26.5−45.8i)T2 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2}
59 1+(−3+5.19i)T+(−29.5−51.0i)T2 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2}
61 1+(2.5+4.33i)T+(−30.5+52.8i)T2 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2}
67 1+(2.5−4.33i)T+(−33.5−58.0i)T2 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2}
71 1+6T+71T2 1 + 6T + 71T^{2}
73 1+(−8+13.8i)T+(−36.5−63.2i)T2 1 + (-8 + 13.8i)T + (-36.5 - 63.2i)T^{2}
79 1+(1+1.73i)T+(−39.5+68.4i)T2 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2}
83 1−3T+83T2 1 - 3T + 83T^{2}
89 1+(−7.5−12.9i)T+(−44.5+77.0i)T2 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2}
97 1−14T+97T2 1 - 14T + 97T^{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

−14.81856775359933444898427806420, −13.95405510801181231302130060545, −13.10628249052049853098496413897, −11.46184298868531629539847871046, −10.50931177147680436079438206836, −9.260788071972860386282139750579, −7.64235446639417735456485637220, −6.51434725552277554544789032162, −5.05512227067677025191579081551, −3.60004486490227968671810948230, 2.06539597472512688526896396217, 4.36637257403942965751068337280, 5.87268498267100909281846045227, 7.20871338231226987004533945840, 9.192176858265344691468648731669, 9.836512944089622846365789091494, 11.66137905510734289213836183759, 12.43727438283056963935294979956, 12.86032123579720894901900042944, 14.64560738992494087806062720888

Graph of the ZZ-function along the critical line