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Properties

Label 2-140-140.79-c0-0-1
Degree 22
Conductor 140140
Sign 0.0633+0.997i0.0633 + 0.997i
Analytic cond. 0.06986910.0698691
Root an. cond. 0.2643270.264327
Motivic weight 00
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 + 0.866i)7-s − 0.999·8-s + (0.499 + 0.866i)10-s + (−0.499 + 0.866i)12-s + 0.999·14-s + 0.999·15-s + (−0.5 + 0.866i)16-s + 0.999·20-s + (0.499 − 0.866i)21-s + (−0.5 + 0.866i)23-s + (0.499 + 0.866i)24-s + ⋯
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 + 0.866i)7-s − 0.999·8-s + (0.499 + 0.866i)10-s + (−0.499 + 0.866i)12-s + 0.999·14-s + 0.999·15-s + (−0.5 + 0.866i)16-s + 0.999·20-s + (0.499 − 0.866i)21-s + (−0.5 + 0.866i)23-s + (0.499 + 0.866i)24-s + ⋯

Functional equation

Λ(s)=(140s/2 ΓC(s) L(s)=((0.0633+0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(140s/2 ΓC(s) L(s)=((0.0633+0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 140140    =    22⋅5⋅72^{2} \cdot 5 \cdot 7
Sign: 0.0633+0.997i0.0633 + 0.997i
Analytic conductor: 0.06986910.0698691
Root analytic conductor: 0.2643270.264327
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ140(79,⋅)\chi_{140} (79, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 140, ( :0), 0.0633+0.997i)(2,\ 140,\ (\ :0),\ 0.0633 + 0.997i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.63801482780.6380148278
L(12)L(\frac12) ≈\approx 0.63801482780.6380148278
L(1)L(1) 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+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good3 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
13 1−T2 1 - T^{2}
17 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
23 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
29 1+T+T2 1 + T + T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
41 1+T+T2 1 + T + T^{2}
43 1−T+T2 1 - T + T^{2}
47 1+(−1+1.73i)T+(−0.5−0.866i)T2 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2}
53 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
67 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1−T+T2 1 - T + T^{2}
89 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
97 1−T2 1 - T^{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

−12.91953457458712685832370021318, −11.89730415182283369245167736241, −11.59390338033710325339197060635, −10.49131899340443749749830472998, −9.186924837552491493319425722300, −7.71848071459985778649623621210, −6.44825492248817334948738652479, −5.42165937945702167287681403977, −3.69435910234721680945246195208, −2.07304643702320871970159871316, 3.98412937626303021390155283110, 4.62317910412855121571655553895, 5.67425283931826133655283997430, 7.29984296400960119785591340565, 8.208014038692492856373607073400, 9.392275456734232582728614569972, 10.67564303383999914924915166349, 11.71208045192582949871815458486, 12.78482575618033582120647928994, 13.72109285787297334621672318409

Graph of the ZZ-function along the critical line