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Properties

Label 2-2160-5.4-c1-0-40
Degree 22
Conductor 21602160
Sign −0.223+0.974i-0.223 + 0.974i
Analytic cond. 17.247617.2476
Root an. cond. 4.153034.15303
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  + (2.17 + 0.5i)5-s − 4.35i·7-s − 4.35·11-s − 4i·17-s + 6·19-s + 2i·23-s + (4.50 + 2.17i)25-s − 7·31-s + (2.17 − 9.50i)35-s − 8.71i·37-s − 8.71·41-s − 8.71i·43-s + 2i·47-s − 12.0·49-s − 3i·53-s + ⋯
L(s)  = 1  + (0.974 + 0.223i)5-s − 1.64i·7-s − 1.31·11-s − 0.970i·17-s + 1.37·19-s + 0.417i·23-s + (0.900 + 0.435i)25-s − 1.25·31-s + (0.368 − 1.60i)35-s − 1.43i·37-s − 1.36·41-s − 1.32i·43-s + 0.291i·47-s − 1.71·49-s − 0.412i·53-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((−0.223+0.974i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s+1/2) L(s)=((−0.223+0.974i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: −0.223+0.974i-0.223 + 0.974i
Analytic conductor: 17.247617.2476
Root analytic conductor: 4.153034.15303
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2160(1729,⋅)\chi_{2160} (1729, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2160, ( :1/2), −0.223+0.974i)(2,\ 2160,\ (\ :1/2),\ -0.223 + 0.974i)

Particular Values

L(1)L(1) ≈\approx 1.6178786711.617878671
L(12)L(\frac12) ≈\approx 1.6178786711.617878671
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
3 1 1
5 1+(−2.17−0.5i)T 1 + (-2.17 - 0.5i)T
good7 1+4.35iT−7T2 1 + 4.35iT - 7T^{2}
11 1+4.35T+11T2 1 + 4.35T + 11T^{2}
13 1−13T2 1 - 13T^{2}
17 1+4iT−17T2 1 + 4iT - 17T^{2}
19 1−6T+19T2 1 - 6T + 19T^{2}
23 1−2iT−23T2 1 - 2iT - 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1+7T+31T2 1 + 7T + 31T^{2}
37 1+8.71iT−37T2 1 + 8.71iT - 37T^{2}
41 1+8.71T+41T2 1 + 8.71T + 41T^{2}
43 1+8.71iT−43T2 1 + 8.71iT - 43T^{2}
47 1−2iT−47T2 1 - 2iT - 47T^{2}
53 1+3iT−53T2 1 + 3iT - 53T^{2}
59 1−8.71T+59T2 1 - 8.71T + 59T^{2}
61 1+4T+61T2 1 + 4T + 61T^{2}
67 1+8.71iT−67T2 1 + 8.71iT - 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+4.35iT−73T2 1 + 4.35iT - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1−5iT−83T2 1 - 5iT - 83T^{2}
89 1−8.71T+89T2 1 - 8.71T + 89T^{2}
97 1+4.35iT−97T2 1 + 4.35iT - 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

−9.061287338584154406616475835429, −7.76800752232354340969992335945, −7.35004437476871214801496744744, −6.70400955932220577077470496976, −5.40009491695299035049301584719, −5.14456478140879043960075343130, −3.81824299601216125343276134828, −2.99464306180405638829342923747, −1.83559711184905792660262808855, −0.54005133197055822449885771424, 1.56666601207922982139473079595, 2.48755056635725276733944794371, 3.20200839323780844130701654935, 4.82326152430334220606221475055, 5.45132019166679165909671016355, 5.88486428964193287050253707565, 6.82973405831949292375592684731, 8.005079568466973848509156008449, 8.534251404105805207297021292054, 9.298046256651684301103594103745

Graph of the ZZ-function along the critical line