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Properties

Label 2-980-140.39-c0-0-1
Degree 22
Conductor 980980
Sign 0.944−0.328i0.944 - 0.328i
Analytic cond. 0.4890830.489083
Root an. cond. 0.6993450.699345
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.965 − 0.258i)5-s + 0.999i·8-s + (0.5 + 0.866i)9-s + (0.965 − 0.258i)10-s − 1.41i·13-s + (−0.5 − 0.866i)16-s + (1.22 + 0.707i)17-s + (−0.866 − 0.499i)18-s + (−0.707 + 0.707i)20-s + (0.866 + 0.499i)25-s + (0.707 + 1.22i)26-s + (0.866 + 0.499i)32-s − 1.41·34-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.965 − 0.258i)5-s + 0.999i·8-s + (0.5 + 0.866i)9-s + (0.965 − 0.258i)10-s − 1.41i·13-s + (−0.5 − 0.866i)16-s + (1.22 + 0.707i)17-s + (−0.866 − 0.499i)18-s + (−0.707 + 0.707i)20-s + (0.866 + 0.499i)25-s + (0.707 + 1.22i)26-s + (0.866 + 0.499i)32-s − 1.41·34-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((0.944−0.328i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s) L(s)=((0.944−0.328i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: 0.944−0.328i0.944 - 0.328i
Analytic conductor: 0.4890830.489083
Root analytic conductor: 0.6993450.699345
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ980(459,⋅)\chi_{980} (459, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :0), 0.944−0.328i)(2,\ 980,\ (\ :0),\ 0.944 - 0.328i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.62157990390.6215799039
L(12)L(\frac12) ≈\approx 0.62157990390.6215799039
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+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
5 1+(0.965+0.258i)T 1 + (0.965 + 0.258i)T
7 1 1
good3 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
13 1+1.41iT−T2 1 + 1.41iT - T^{2}
17 1+(−1.22−0.707i)T+(0.5+0.866i)T2 1 + (-1.22 - 0.707i)T + (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)T2 1 + (-0.5 + 0.866i)T^{2}
29 1+T2 1 + T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1+(−1.73+i)T+(0.5−0.866i)T2 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2}
41 1−1.41T+T2 1 - 1.41T + T^{2}
43 1+T2 1 + T^{2}
47 1+(−0.5+0.866i)T2 1 + (-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+(−0.707−1.22i)T+(−0.5+0.866i)T2 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2}
67 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+(1.22+0.707i)T+(0.5+0.866i)T2 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2}
79 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1+T2 1 + T^{2}
89 1+(0.707+1.22i)T+(−0.5+0.866i)T2 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2}
97 1−1.41iT−T2 1 - 1.41iT - 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

−10.31039097304477237609774471422, −9.350958932551236198169326493779, −8.299000091668864983571778507251, −7.77847901556412539578274787496, −7.35254388748331778906011670772, −5.97175471216079513764284289979, −5.24837717668583242438779797325, −4.11615482734074465580480162201, −2.72931873858675939607097594529, −1.09103090404951139878806406777, 1.12503455848332174329274823098, 2.73960846638942985615717002133, 3.74268858308021409640504315515, 4.48927676494909329218206359569, 6.25535783319026214325310758899, 7.06652642881219069541522391606, 7.67021436107121208520001598496, 8.583758314404481453244171368113, 9.493600949540969785583798795577, 9.922552197709276684957316387725

Graph of the ZZ-function along the critical line