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Properties

Label 2-84-21.17-c1-0-0
Degree 22
Conductor 8484
Sign 0.553−0.832i0.553 - 0.832i
Analytic cond. 0.6707430.670743
Root an. cond. 0.8189890.818989
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  + (−1.5 + 0.866i)3-s + (1.5 + 2.59i)5-s + (2 + 1.73i)7-s + (1.5 − 2.59i)9-s + (−4.5 − 2.59i)11-s + (−4.5 − 2.59i)15-s + (1.5 − 2.59i)17-s + (1.5 − 0.866i)19-s + (−4.5 − 0.866i)21-s + (4.5 − 2.59i)23-s + (−2 + 3.46i)25-s + 5.19i·27-s + (−1.5 − 0.866i)31-s + 9·33-s + (−1.5 + 7.79i)35-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)3-s + (0.670 + 1.16i)5-s + (0.755 + 0.654i)7-s + (0.5 − 0.866i)9-s + (−1.35 − 0.783i)11-s + (−1.16 − 0.670i)15-s + (0.363 − 0.630i)17-s + (0.344 − 0.198i)19-s + (−0.981 − 0.188i)21-s + (0.938 − 0.541i)23-s + (−0.400 + 0.692i)25-s + 0.999i·27-s + (−0.269 − 0.155i)31-s + 1.56·33-s + (−0.253 + 1.31i)35-s + ⋯

Functional equation

Λ(s)=(84s/2 ΓC(s) L(s)=((0.553−0.832i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(84s/2 ΓC(s+1/2) L(s)=((0.553−0.832i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8484    =    22⋅3⋅72^{2} \cdot 3 \cdot 7
Sign: 0.553−0.832i0.553 - 0.832i
Analytic conductor: 0.6707430.670743
Root analytic conductor: 0.8189890.818989
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ84(17,⋅)\chi_{84} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 84, ( :1/2), 0.553−0.832i)(2,\ 84,\ (\ :1/2),\ 0.553 - 0.832i)

Particular Values

L(1)L(1) ≈\approx 0.748335+0.400992i0.748335 + 0.400992i
L(12)L(\frac12) ≈\approx 0.748335+0.400992i0.748335 + 0.400992i
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+(1.5−0.866i)T 1 + (1.5 - 0.866i)T
7 1+(−2−1.73i)T 1 + (-2 - 1.73i)T
good5 1+(−1.5−2.59i)T+(−2.5+4.33i)T2 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2}
11 1+(4.5+2.59i)T+(5.5+9.52i)T2 1 + (4.5 + 2.59i)T + (5.5 + 9.52i)T^{2}
13 1−13T2 1 - 13T^{2}
17 1+(−1.5+2.59i)T+(−8.5−14.7i)T2 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.5+0.866i)T+(9.5−16.4i)T2 1 + (-1.5 + 0.866i)T + (9.5 - 16.4i)T^{2}
23 1+(−4.5+2.59i)T+(11.5−19.9i)T2 1 + (-4.5 + 2.59i)T + (11.5 - 19.9i)T^{2}
29 1−29T2 1 - 29T^{2}
31 1+(1.5+0.866i)T+(15.5+26.8i)T2 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2}
37 1+(3.5+6.06i)T+(−18.5+32.0i)T2 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1−4T+43T2 1 - 4T + 43T^{2}
47 1+(−1.5−2.59i)T+(−23.5+40.7i)T2 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2}
53 1+(−4.5−2.59i)T+(26.5+45.8i)T2 1 + (-4.5 - 2.59i)T + (26.5 + 45.8i)T^{2}
59 1+(1.5−2.59i)T+(−29.5−51.0i)T2 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2}
61 1+(10.5−6.06i)T+(30.5−52.8i)T2 1 + (10.5 - 6.06i)T + (30.5 - 52.8i)T^{2}
67 1+(2.5−4.33i)T+(−33.5−58.0i)T2 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2}
71 1+10.3iT−71T2 1 + 10.3iT - 71T^{2}
73 1+(10.5+6.06i)T+(36.5+63.2i)T2 1 + (10.5 + 6.06i)T + (36.5 + 63.2i)T^{2}
79 1+(−0.5−0.866i)T+(−39.5+68.4i)T2 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2}
83 1−12T+83T2 1 - 12T + 83T^{2}
89 1+(4.5+7.79i)T+(−44.5+77.0i)T2 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2}
97 1−6.92iT−97T2 1 - 6.92iT - 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

−14.59174071179486161604584992940, −13.47331276317274255234512181947, −12.05639591848514170888361517975, −10.92505200911177285641596375675, −10.47296983286332629393831550726, −9.075181555363793739476404870973, −7.39495330944302374714503174829, −5.97543486726493630289169416536, −5.09949887049444236650262658522, −2.85316685493529774338497922753, 1.54619984643980010301992483740, 4.79948549506521339838917428403, 5.48565986050273276885501079767, 7.22703844650902517333582839116, 8.277117826161875237870392633277, 9.918395853026315812525942744810, 10.83467187824897371894014100415, 12.16075189994791473413079363820, 13.01658424383537857920447951262, 13.69169966142622659909948882410

Graph of the ZZ-function along the critical line