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Properties

Label 2-4600-5.4-c1-0-95
Degree 22
Conductor 46004600
Sign −0.447−0.894i-0.447 - 0.894i
Analytic cond. 36.731136.7311
Root an. cond. 6.060626.06062
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  − 1.93i·3-s − 4.93i·7-s − 0.745·9-s + 0.745·11-s + 1.74i·13-s + 6.10i·17-s − 5.44·19-s − 9.55·21-s − i·23-s − 4.36i·27-s + 1.66·29-s − 1.61·31-s − 1.44i·33-s − 4.34i·37-s + 3.37·39-s + ⋯
L(s)  = 1  − 1.11i·3-s − 1.86i·7-s − 0.248·9-s + 0.224·11-s + 0.484i·13-s + 1.48i·17-s − 1.24·19-s − 2.08·21-s − 0.208i·23-s − 0.839i·27-s + 0.309·29-s − 0.290·31-s − 0.251i·33-s − 0.714i·37-s + 0.541·39-s + ⋯

Functional equation

Λ(s)=(4600s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4600s/2 ΓC(s+1/2) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 46004600    =    23⋅52⋅232^{3} \cdot 5^{2} \cdot 23
Sign: −0.447−0.894i-0.447 - 0.894i
Analytic conductor: 36.731136.7311
Root analytic conductor: 6.060626.06062
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4600(4049,⋅)\chi_{4600} (4049, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4600, ( :1/2), −0.447−0.894i)(2,\ 4600,\ (\ :1/2),\ -0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 0.47601786890.4760178689
L(12)L(\frac12) ≈\approx 0.47601786890.4760178689
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
5 1 1
23 1+iT 1 + iT
good3 1+1.93iT−3T2 1 + 1.93iT - 3T^{2}
7 1+4.93iT−7T2 1 + 4.93iT - 7T^{2}
11 1−0.745T+11T2 1 - 0.745T + 11T^{2}
13 1−1.74iT−13T2 1 - 1.74iT - 13T^{2}
17 1−6.10iT−17T2 1 - 6.10iT - 17T^{2}
19 1+5.44T+19T2 1 + 5.44T + 19T^{2}
29 1−1.66T+29T2 1 - 1.66T + 29T^{2}
31 1+1.61T+31T2 1 + 1.61T + 31T^{2}
37 1+4.34iT−37T2 1 + 4.34iT - 37T^{2}
41 1+6.95T+41T2 1 + 6.95T + 41T^{2}
43 1−5.01iT−43T2 1 - 5.01iT - 43T^{2}
47 1+2.68iT−47T2 1 + 2.68iT - 47T^{2}
53 1−13.7iT−53T2 1 - 13.7iT - 53T^{2}
59 1+12.2T+59T2 1 + 12.2T + 59T^{2}
61 1+13.9T+61T2 1 + 13.9T + 61T^{2}
67 1+13.1iT−67T2 1 + 13.1iT - 67T^{2}
71 1+9.67T+71T2 1 + 9.67T + 71T^{2}
73 1−5.69iT−73T2 1 - 5.69iT - 73T^{2}
79 1+10.3T+79T2 1 + 10.3T + 79T^{2}
83 1−0.637iT−83T2 1 - 0.637iT - 83T^{2}
89 1+2.72T+89T2 1 + 2.72T + 89T^{2}
97 1+7.12iT−97T2 1 + 7.12iT - 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

−7.64510109453639092444884024778, −7.19271016416150135325467800361, −6.40705381377134358207069369889, −6.11274483258941312505808345948, −4.47095096408312255648799581633, −4.24226064313106523390457887425, −3.22685803951788473370190911113, −1.83047113678548038974272531737, −1.35162024356850711900845100528, −0.12489895260359317890238115803, 1.79603649955825930895827371337, 2.78180705870331648111531087966, 3.38626602287232560670770665231, 4.57021592440584243123759162702, 5.01153668558753905536227323973, 5.73445496213636607578610493764, 6.45469418068813699109999749919, 7.39211620806571498592970133249, 8.447158371472649501837890978148, 8.852725181981246116838660163508

Graph of the ZZ-function along the critical line