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Properties

Label 2-975-5.4-c1-0-23
Degree 22
Conductor 975975
Sign 0.894+0.447i0.894 + 0.447i
Analytic cond. 7.785417.78541
Root an. cond. 2.790232.79023
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 − i·3-s + 4-s + 6-s − 3i·7-s + 3i·8-s − 9-s − 11-s − i·12-s − i·13-s + 3·14-s − 16-s − 5i·17-s − i·18-s + 8·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s + 0.5·4-s + 0.408·6-s − 1.13i·7-s + 1.06i·8-s − 0.333·9-s − 0.301·11-s − 0.288i·12-s − 0.277i·13-s + 0.801·14-s − 0.250·16-s − 1.21i·17-s − 0.235i·18-s + 1.83·19-s + ⋯

Functional equation

Λ(s)=(975s/2 ΓC(s) L(s)=((0.894+0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(975s/2 ΓC(s+1/2) L(s)=((0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{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: 975975    =    3⋅52⋅133 \cdot 5^{2} \cdot 13
Sign: 0.894+0.447i0.894 + 0.447i
Analytic conductor: 7.785417.78541
Root analytic conductor: 2.790232.79023
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ975(274,⋅)\chi_{975} (274, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 975, ( :1/2), 0.894+0.447i)(2,\ 975,\ (\ :1/2),\ 0.894 + 0.447i)

Particular Values

L(1)L(1) ≈\approx 1.79410−0.423531i1.79410 - 0.423531i
L(12)L(\frac12) ≈\approx 1.79410−0.423531i1.79410 - 0.423531i
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)
bad3 1+iT 1 + iT
5 1 1
13 1+iT 1 + iT
good2 1−iT−2T2 1 - iT - 2T^{2}
7 1+3iT−7T2 1 + 3iT - 7T^{2}
11 1+T+11T2 1 + T + 11T^{2}
17 1+5iT−17T2 1 + 5iT - 17T^{2}
19 1−8T+19T2 1 - 8T + 19T^{2}
23 1−23T2 1 - 23T^{2}
29 1+T+29T2 1 + T + 29T^{2}
31 1−3T+31T2 1 - 3T + 31T^{2}
37 1+8iT−37T2 1 + 8iT - 37T^{2}
41 1+2T+41T2 1 + 2T + 41T^{2}
43 1+8iT−43T2 1 + 8iT - 43T^{2}
47 1+11iT−47T2 1 + 11iT - 47T^{2}
53 1−11iT−53T2 1 - 11iT - 53T^{2}
59 1+5T+59T2 1 + 5T + 59T^{2}
61 1−T+61T2 1 - T + 61T^{2}
67 1−3iT−67T2 1 - 3iT - 67T^{2}
71 1−16T+71T2 1 - 16T + 71T^{2}
73 1−4iT−73T2 1 - 4iT - 73T^{2}
79 1+12T+79T2 1 + 12T + 79T^{2}
83 1−3iT−83T2 1 - 3iT - 83T^{2}
89 1+89T2 1 + 89T^{2}
97 1+2iT−97T2 1 + 2iT - 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

−9.993365486017427421181187407919, −8.931870849238938903462290077281, −7.81579865387732758228568223813, −7.34426558003431121544280184614, −6.88062157319066545505867244925, −5.70509225015289243159030107494, −5.02701032876611107425291241719, −3.52997842255030820555157743329, −2.42895203304094878911833113373, −0.902528051191669292261780888325, 1.53357364983604927373442074658, 2.74840290685116707800225510351, 3.44831572882204286425184494551, 4.72309955422586542185131622997, 5.74135727289181725149398634475, 6.48066467962313670565829759637, 7.69713105689630168693199699420, 8.548589806551599059976766094615, 9.610572909105518852360956890047, 9.953219325022572253019619379447

Graph of the ZZ-function along the critical line