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Properties

Label 2-1872-12.11-c1-0-18
Degree 22
Conductor 18721872
Sign −0.418+0.908i-0.418 + 0.908i
Analytic cond. 14.947914.9479
Root an. cond. 3.866263.86626
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  − 1.41i·5-s + 0.504i·7-s − 3.16·11-s + 13-s + 5.47i·17-s − 8.44i·19-s + 0.712·23-s + 2.99·25-s − 4.24i·29-s − 2.96i·31-s + 0.712·35-s − 5.74·37-s − 9.54i·41-s − 3.46i·43-s − 12.9·47-s + ⋯
L(s)  = 1  − 0.632i·5-s + 0.190i·7-s − 0.953·11-s + 0.277·13-s + 1.32i·17-s − 1.93i·19-s + 0.148·23-s + 0.599·25-s − 0.787i·29-s − 0.531i·31-s + 0.120·35-s − 0.944·37-s − 1.48i·41-s − 0.528i·43-s − 1.89·47-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=((−0.418+0.908i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s+1/2) L(s)=((−0.418+0.908i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: −0.418+0.908i-0.418 + 0.908i
Analytic conductor: 14.947914.9479
Root analytic conductor: 3.866263.86626
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1872(287,⋅)\chi_{1872} (287, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1872, ( :1/2), −0.418+0.908i)(2,\ 1872,\ (\ :1/2),\ -0.418 + 0.908i)

Particular Values

L(1)L(1) ≈\approx 1.0810883421.081088342
L(12)L(\frac12) ≈\approx 1.0810883421.081088342
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
13 1−T 1 - T
good5 1+1.41iT−5T2 1 + 1.41iT - 5T^{2}
7 1−0.504iT−7T2 1 - 0.504iT - 7T^{2}
11 1+3.16T+11T2 1 + 3.16T + 11T^{2}
17 1−5.47iT−17T2 1 - 5.47iT - 17T^{2}
19 1+8.44iT−19T2 1 + 8.44iT - 19T^{2}
23 1−0.712T+23T2 1 - 0.712T + 23T^{2}
29 1+4.24iT−29T2 1 + 4.24iT - 29T^{2}
31 1+2.96iT−31T2 1 + 2.96iT - 31T^{2}
37 1+5.74T+37T2 1 + 5.74T + 37T^{2}
41 1+9.54iT−41T2 1 + 9.54iT - 41T^{2}
43 1+3.46iT−43T2 1 + 3.46iT - 43T^{2}
47 1+12.9T+47T2 1 + 12.9T + 47T^{2}
53 1+8.30iT−53T2 1 + 8.30iT - 53T^{2}
59 1+7.34T+59T2 1 + 7.34T + 59T^{2}
61 1−7.74T+61T2 1 - 7.74T + 61T^{2}
67 1−4.97iT−67T2 1 - 4.97iT - 67T^{2}
71 1+1.02T+71T2 1 + 1.02T + 71T^{2}
73 1+9.74T+73T2 1 + 9.74T + 73T^{2}
79 1+7.93iT−79T2 1 + 7.93iT - 79T^{2}
83 1+7.34T+83T2 1 + 7.34T + 83T^{2}
89 1−9.54iT−89T2 1 - 9.54iT - 89T^{2}
97 1+1.74T+97T2 1 + 1.74T + 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

−8.709037339052520537815242663314, −8.494231537180586699436114327336, −7.42779855253820524501948181148, −6.62686647037929914847803560343, −5.61975364682484081873438670682, −4.97669214695906394163000787648, −4.08668614249884787066353327614, −2.93417566780936980139582401814, −1.88959518926525598102624764829, −0.39586876152838468491542788581, 1.42430272825997964298113751405, 2.80011271893989306240606000665, 3.41440422449919638771454757484, 4.66516137410667902768469309259, 5.43936860763251445759935024676, 6.36936759803655462531520299787, 7.16663300389046745624264358855, 7.85054463799691191775073413400, 8.604223775571914117814651664216, 9.619483488354640315752156638497

Graph of the ZZ-function along the critical line