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Properties

Label 2-768-8.5-c1-0-7
Degree 22
Conductor 768768
Sign 0.707+0.707i0.707 + 0.707i
Analytic cond. 6.132516.13251
Root an. cond. 2.476392.47639
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  − i·3-s + 2i·5-s − 9-s − 4i·11-s − 2i·13-s + 2·15-s + 2·17-s − 4i·19-s + 8·23-s + 25-s + i·27-s + 6i·29-s + 8·31-s − 4·33-s − 6i·37-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.894i·5-s − 0.333·9-s − 1.20i·11-s − 0.554i·13-s + 0.516·15-s + 0.485·17-s − 0.917i·19-s + 1.66·23-s + 0.200·25-s + 0.192i·27-s + 1.11i·29-s + 1.43·31-s − 0.696·33-s − 0.986i·37-s + ⋯

Functional equation

Λ(s)=(768s/2 ΓC(s) L(s)=((0.707+0.707i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(768s/2 ΓC(s+1/2) L(s)=((0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 768768    =    28⋅32^{8} \cdot 3
Sign: 0.707+0.707i0.707 + 0.707i
Analytic conductor: 6.132516.13251
Root analytic conductor: 2.476392.47639
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ768(385,⋅)\chi_{768} (385, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 768, ( :1/2), 0.707+0.707i)(2,\ 768,\ (\ :1/2),\ 0.707 + 0.707i)

Particular Values

L(1)L(1) ≈\approx 1.40881−0.583548i1.40881 - 0.583548i
L(12)L(\frac12) ≈\approx 1.40881−0.583548i1.40881 - 0.583548i
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+iT 1 + iT
good5 1−2iT−5T2 1 - 2iT - 5T^{2}
7 1+7T2 1 + 7T^{2}
11 1+4iT−11T2 1 + 4iT - 11T^{2}
13 1+2iT−13T2 1 + 2iT - 13T^{2}
17 1−2T+17T2 1 - 2T + 17T^{2}
19 1+4iT−19T2 1 + 4iT - 19T^{2}
23 1−8T+23T2 1 - 8T + 23T^{2}
29 1−6iT−29T2 1 - 6iT - 29T^{2}
31 1−8T+31T2 1 - 8T + 31T^{2}
37 1+6iT−37T2 1 + 6iT - 37T^{2}
41 1−6T+41T2 1 - 6T + 41T^{2}
43 1+4iT−43T2 1 + 4iT - 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1−2iT−53T2 1 - 2iT - 53T^{2}
59 1+4iT−59T2 1 + 4iT - 59T^{2}
61 1+2iT−61T2 1 + 2iT - 61T^{2}
67 1+4iT−67T2 1 + 4iT - 67T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1+10T+73T2 1 + 10T + 73T^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1+4iT−83T2 1 + 4iT - 83T^{2}
89 1−6T+89T2 1 - 6T + 89T^{2}
97 1−2T+97T2 1 - 2T + 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

−10.53678054538526819429902130252, −9.221881939788189024996817046142, −8.488160237866709217936228452879, −7.48771610208422705976173381661, −6.79040057983196740369841487635, −5.94958079517156289184029044613, −4.95170658338410290830759369963, −3.29429813718658719386393794195, −2.75410683407352004343693392806, −0.912968318039718484977089011385, 1.34453381360352526103933658464, 2.89013753384545291997686266012, 4.34684880364153666893216696261, 4.76249930341247046819410826981, 5.87174871793317313828757853650, 6.98073821428419389803845495304, 8.009929251330985480190217384238, 8.819304113030716990485867333560, 9.677381846222648336617337407700, 10.14143597954171516893615931924

Graph of the ZZ-function along the critical line