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Properties

Label 2-4680-5.4-c1-0-40
Degree 22
Conductor 46804680
Sign 0.590−0.807i0.590 - 0.807i
Analytic cond. 37.369937.3699
Root an. cond. 6.113096.11309
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  + (−1.32 + 1.80i)5-s + 4.64·11-s + i·13-s + 4.24i·17-s + 6.24·19-s − 2.24i·23-s + (−1.51 − 4.76i)25-s − 9.21·29-s + 9.28·31-s − 7.28i·37-s + 5.67·41-s − 4.24i·43-s + 2.88i·47-s + 7·49-s + 9.21i·53-s + ⋯
L(s)  = 1  + (−0.590 + 0.807i)5-s + 1.39·11-s + 0.277i·13-s + 1.03i·17-s + 1.43·19-s − 0.469i·23-s + (−0.303 − 0.952i)25-s − 1.71·29-s + 1.66·31-s − 1.19i·37-s + 0.885·41-s − 0.648i·43-s + 0.421i·47-s + 49-s + 1.26i·53-s + ⋯

Functional equation

Λ(s)=(4680s/2 ΓC(s) L(s)=((0.590−0.807i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4680s/2 ΓC(s+1/2) L(s)=((0.590−0.807i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 46804680    =    23⋅32⋅5⋅132^{3} \cdot 3^{2} \cdot 5 \cdot 13
Sign: 0.590−0.807i0.590 - 0.807i
Analytic conductor: 37.369937.3699
Root analytic conductor: 6.113096.11309
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4680(2809,⋅)\chi_{4680} (2809, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4680, ( :1/2), 0.590−0.807i)(2,\ 4680,\ (\ :1/2),\ 0.590 - 0.807i)

Particular Values

L(1)L(1) ≈\approx 1.9358770961.935877096
L(12)L(\frac12) ≈\approx 1.9358770961.935877096
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 1+(1.32−1.80i)T 1 + (1.32 - 1.80i)T
13 1−iT 1 - iT
good7 1−7T2 1 - 7T^{2}
11 1−4.64T+11T2 1 - 4.64T + 11T^{2}
17 1−4.24iT−17T2 1 - 4.24iT - 17T^{2}
19 1−6.24T+19T2 1 - 6.24T + 19T^{2}
23 1+2.24iT−23T2 1 + 2.24iT - 23T^{2}
29 1+9.21T+29T2 1 + 9.21T + 29T^{2}
31 1−9.28T+31T2 1 - 9.28T + 31T^{2}
37 1+7.28iT−37T2 1 + 7.28iT - 37T^{2}
41 1−5.67T+41T2 1 - 5.67T + 41T^{2}
43 1+4.24iT−43T2 1 + 4.24iT - 43T^{2}
47 1−2.88iT−47T2 1 - 2.88iT - 47T^{2}
53 1−9.21iT−53T2 1 - 9.21iT - 53T^{2}
59 1−5.92T+59T2 1 - 5.92T + 59T^{2}
61 1−0.969T+61T2 1 - 0.969T + 61T^{2}
67 1−1.93iT−67T2 1 - 1.93iT - 67T^{2}
71 1+5.60T+71T2 1 + 5.60T + 71T^{2}
73 1+12.5iT−73T2 1 + 12.5iT - 73T^{2}
79 1+12.2T+79T2 1 + 12.2T + 79T^{2}
83 1−3.67iT−83T2 1 - 3.67iT - 83T^{2}
89 1+9.67T+89T2 1 + 9.67T + 89T^{2}
97 1−6iT−97T2 1 - 6iT - 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

−8.411724813109780973455536093848, −7.46222732410433249347188830150, −7.14800034352124969813701017431, −6.20086024720175438117583436428, −5.74500155277341439032436228531, −4.39989038625850932156593794582, −3.91448958986992446456638781821, −3.16240868630571458552652351563, −2.10553046701344435642962485854, −0.948629858708896455709054543124, 0.71982684269182858581747500371, 1.48852049399851799998283749392, 2.91116198848940890657210184104, 3.71361675055774189050791387533, 4.43745917766107825015043634404, 5.22374879187697506366668300811, 5.89183448816922971218214625537, 6.95005375838309732910075642872, 7.43208548335375710538221930372, 8.230697021696547120823972166986

Graph of the ZZ-function along the critical line