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Properties

Label 2-968-8.5-c1-0-74
Degree 22
Conductor 968968
Sign 0.781+0.623i0.781 + 0.623i
Analytic cond. 7.729517.72951
Root an. cond. 2.780202.78020
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  + (1.37 + 0.314i)2-s − 1.87i·3-s + (1.80 + 0.867i)4-s + 0.493i·5-s + (0.591 − 2.59i)6-s + 0.174·7-s + (2.21 + 1.76i)8-s − 0.534·9-s + (−0.155 + 0.679i)10-s + (1.63 − 3.38i)12-s − 5.51i·13-s + (0.240 + 0.0549i)14-s + 0.927·15-s + (2.49 + 3.12i)16-s − 0.678·17-s + (−0.736 − 0.168i)18-s + ⋯
L(s)  = 1  + (0.974 + 0.222i)2-s − 1.08i·3-s + (0.900 + 0.433i)4-s + 0.220i·5-s + (0.241 − 1.05i)6-s + 0.0660·7-s + (0.781 + 0.623i)8-s − 0.178·9-s + (−0.0490 + 0.214i)10-s + (0.470 − 0.977i)12-s − 1.53i·13-s + (0.0644 + 0.0146i)14-s + 0.239·15-s + (0.623 + 0.781i)16-s − 0.164·17-s + (−0.173 − 0.0396i)18-s + ⋯

Functional equation

Λ(s)=(968s/2 ΓC(s) L(s)=((0.781+0.623i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(968s/2 ΓC(s+1/2) L(s)=((0.781+0.623i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 968968    =    23⋅1122^{3} \cdot 11^{2}
Sign: 0.781+0.623i0.781 + 0.623i
Analytic conductor: 7.729517.72951
Root analytic conductor: 2.780202.78020
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ968(485,⋅)\chi_{968} (485, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 968, ( :1/2), 0.781+0.623i)(2,\ 968,\ (\ :1/2),\ 0.781 + 0.623i)

Particular Values

L(1)L(1) ≈\approx 2.96202−1.03633i2.96202 - 1.03633i
L(12)L(\frac12) ≈\approx 2.96202−1.03633i2.96202 - 1.03633i
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+(−1.37−0.314i)T 1 + (-1.37 - 0.314i)T
11 1 1
good3 1+1.87iT−3T2 1 + 1.87iT - 3T^{2}
5 1−0.493iT−5T2 1 - 0.493iT - 5T^{2}
7 1−0.174T+7T2 1 - 0.174T + 7T^{2}
13 1+5.51iT−13T2 1 + 5.51iT - 13T^{2}
17 1+0.678T+17T2 1 + 0.678T + 17T^{2}
19 1−0.437iT−19T2 1 - 0.437iT - 19T^{2}
23 1−6.88T+23T2 1 - 6.88T + 23T^{2}
29 1−1.41iT−29T2 1 - 1.41iT - 29T^{2}
31 1+4.17T+31T2 1 + 4.17T + 31T^{2}
37 1−6.52iT−37T2 1 - 6.52iT - 37T^{2}
41 1+9.78T+41T2 1 + 9.78T + 41T^{2}
43 1+8.22iT−43T2 1 + 8.22iT - 43T^{2}
47 1−7.75T+47T2 1 - 7.75T + 47T^{2}
53 1−9.91iT−53T2 1 - 9.91iT - 53T^{2}
59 1+9.45iT−59T2 1 + 9.45iT - 59T^{2}
61 1−9.60iT−61T2 1 - 9.60iT - 61T^{2}
67 1+5.24iT−67T2 1 + 5.24iT - 67T^{2}
71 1−4.61T+71T2 1 - 4.61T + 71T^{2}
73 1+6.85T+73T2 1 + 6.85T + 73T^{2}
79 1+6.77T+79T2 1 + 6.77T + 79T^{2}
83 1−7.28iT−83T2 1 - 7.28iT - 83T^{2}
89 1+9.61T+89T2 1 + 9.61T + 89T^{2}
97 1−3.35T+97T2 1 - 3.35T + 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

−10.28071517965207541241693706936, −8.758574808239847694797530786248, −7.919270338354706520092549661030, −7.17222617322004455602549030534, −6.63464133362946486754019910229, −5.61076259342448570368284218468, −4.84147933377345988747747312986, −3.43692019599691049251092601791, −2.60464750423640798471752386560, −1.26001895355764569819536841474, 1.64239094254195054810400078110, 3.04143979717963280657257944873, 4.04771529307581140014110776940, 4.68611493365299725731799295830, 5.38185629742278033600226171642, 6.63047710731879886484650932140, 7.23836636624702271554847000713, 8.749424306163128444719080453125, 9.412114372747369386056249802151, 10.21965636984578522824572563519

Graph of the ZZ-function along the critical line