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Properties

Label 2-980-5.4-c1-0-6
Degree 22
Conductor 980980
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 7.825337.82533
Root an. cond. 2.797382.79738
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  + (−1 + 2i)5-s + 3·9-s − 4i·13-s + 4i·17-s + 4·19-s + 8i·23-s + (−3 − 4i)25-s − 2·29-s + 8·31-s + 8i·37-s − 6·41-s + 8i·43-s + (−3 + 6i)45-s + 8i·47-s − 4·59-s + ⋯
L(s)  = 1  + (−0.447 + 0.894i)5-s + 9-s − 1.10i·13-s + 0.970i·17-s + 0.917·19-s + 1.66i·23-s + (−0.600 − 0.800i)25-s − 0.371·29-s + 1.43·31-s + 1.31i·37-s − 0.937·41-s + 1.21i·43-s + (−0.447 + 0.894i)45-s + 1.16i·47-s − 0.520·59-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(589,⋅)\chi_{980} (589, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), 0.447−0.894i)(2,\ 980,\ (\ :1/2),\ 0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.28325+0.793098i1.28325 + 0.793098i
L(12)L(\frac12) ≈\approx 1.28325+0.793098i1.28325 + 0.793098i
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 1
5 1+(1−2i)T 1 + (1 - 2i)T
7 1 1
good3 1−3T2 1 - 3T^{2}
11 1+11T2 1 + 11T^{2}
13 1+4iT−13T2 1 + 4iT - 13T^{2}
17 1−4iT−17T2 1 - 4iT - 17T^{2}
19 1−4T+19T2 1 - 4T + 19T^{2}
23 1−8iT−23T2 1 - 8iT - 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1−8T+31T2 1 - 8T + 31T^{2}
37 1−8iT−37T2 1 - 8iT - 37T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1−8iT−43T2 1 - 8iT - 43T^{2}
47 1−8iT−47T2 1 - 8iT - 47T^{2}
53 1−53T2 1 - 53T^{2}
59 1+4T+59T2 1 + 4T + 59T^{2}
61 1−6T+61T2 1 - 6T + 61T^{2}
67 1+8iT−67T2 1 + 8iT - 67T^{2}
71 1−12T+71T2 1 - 12T + 71T^{2}
73 1−4iT−73T2 1 - 4iT - 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1−83T2 1 - 83T^{2}
89 1+10T+89T2 1 + 10T + 89T^{2}
97 1+12iT−97T2 1 + 12iT - 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

−10.05233075025307437457255620931, −9.664454064208399134561197183584, −8.142734267157461736751636478283, −7.73245463829029371021455980558, −6.82286378000839713123443864296, −5.97249057170617375311967569198, −4.85222644296536460126174026308, −3.71245667115707378202967247638, −2.97490547179958909482585564847, −1.38123482634235965320934201771, 0.795800109119008986207509747722, 2.17543261314394075770418611888, 3.75797069881104069637612385180, 4.55558651605682777273395501259, 5.27094420487273348104079897818, 6.66267580602216179562967404843, 7.26938529790489518941486150390, 8.257272721551026313630727910791, 9.064975301251896972903238645405, 9.718041539332464626372441339389

Graph of the ZZ-function along the critical line