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Properties

Label 2-1350-5.4-c1-0-2
Degree 22
Conductor 13501350
Sign −0.894−0.447i-0.894 - 0.447i
Analytic cond. 10.779810.7798
Root an. cond. 3.283263.28326
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·2-s − 4-s + i·7-s − i·8-s + 2i·13-s − 14-s + 16-s + 6i·17-s + 19-s − 6i·23-s − 2·26-s − i·28-s − 6·29-s + 5·31-s + i·32-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.377i·7-s − 0.353i·8-s + 0.554i·13-s − 0.267·14-s + 0.250·16-s + 1.45i·17-s + 0.229·19-s − 1.25i·23-s − 0.392·26-s − 0.188i·28-s − 1.11·29-s + 0.898·31-s + 0.176i·32-s + ⋯

Functional equation

Λ(s)=(1350s/2 ΓC(s) L(s)=((−0.894−0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1350s/2 ΓC(s+1/2) L(s)=((−0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13501350    =    2⋅33⋅522 \cdot 3^{3} \cdot 5^{2}
Sign: −0.894−0.447i-0.894 - 0.447i
Analytic conductor: 10.779810.7798
Root analytic conductor: 3.283263.28326
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1350(649,⋅)\chi_{1350} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1350, ( :1/2), −0.894−0.447i)(2,\ 1350,\ (\ :1/2),\ -0.894 - 0.447i)

Particular Values

L(1)L(1) ≈\approx 1.1023538841.102353884
L(12)L(\frac12) ≈\approx 1.1023538841.102353884
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−iT 1 - iT
3 1 1
5 1 1
good7 1−iT−7T2 1 - iT - 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1−2iT−13T2 1 - 2iT - 13T^{2}
17 1−6iT−17T2 1 - 6iT - 17T^{2}
19 1−T+19T2 1 - T + 19T^{2}
23 1+6iT−23T2 1 + 6iT - 23T^{2}
29 1+6T+29T2 1 + 6T + 29T^{2}
31 1−5T+31T2 1 - 5T + 31T^{2}
37 1−7iT−37T2 1 - 7iT - 37T^{2}
41 1+12T+41T2 1 + 12T + 41T^{2}
43 1−11iT−43T2 1 - 11iT - 43T^{2}
47 1−12iT−47T2 1 - 12iT - 47T^{2}
53 1−53T2 1 - 53T^{2}
59 1+6T+59T2 1 + 6T + 59T^{2}
61 1+7T+61T2 1 + 7T + 61T^{2}
67 1−4iT−67T2 1 - 4iT - 67T^{2}
71 1+6T+71T2 1 + 6T + 71T^{2}
73 1+7iT−73T2 1 + 7iT - 73T^{2}
79 1−T+79T2 1 - T + 79T^{2}
83 1−6iT−83T2 1 - 6iT - 83T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1+5iT−97T2 1 + 5iT - 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

−9.836622120398146107072901775391, −8.944114795050053596617946401790, −8.325396380328993940725518319958, −7.57982628729479529230328223731, −6.44522956847503204492409833762, −6.10652183940650925047503430102, −4.91882461052074495254691884640, −4.17832460918002345553434728074, −2.98293020204661445779113799761, −1.55390017061958050955214855550, 0.45994623988403635555091252231, 1.88115637931122366117318753030, 3.09075971304660146306979084986, 3.86161969296874436345554460870, 5.03583953417591464402076865678, 5.64457833596253544159827872758, 7.03961768579907224725114094771, 7.58191430796751067552709408838, 8.659931308072713457773860968380, 9.389517595187708570774154348231

Graph of the ZZ-function along the critical line