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Properties

Label 2-4050-5.4-c1-0-11
Degree 22
Conductor 40504050
Sign −0.894+0.447i-0.894 + 0.447i
Analytic cond. 32.339432.3394
Root an. cond. 5.686775.68677
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 + 4i·7-s − i·8-s − i·13-s − 4·14-s + 16-s + 3i·17-s + 4·19-s + 26-s − 4i·28-s − 9·29-s − 4·31-s + i·32-s − 3·34-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 1.51i·7-s − 0.353i·8-s − 0.277i·13-s − 1.06·14-s + 0.250·16-s + 0.727i·17-s + 0.917·19-s + 0.196·26-s − 0.755i·28-s − 1.67·29-s − 0.718·31-s + 0.176i·32-s − 0.514·34-s + ⋯

Functional equation

Λ(s)=(4050s/2 ΓC(s) L(s)=((−0.894+0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4050s/2 ΓC(s+1/2) L(s)=((−0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{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: 40504050    =    2⋅34⋅522 \cdot 3^{4} \cdot 5^{2}
Sign: −0.894+0.447i-0.894 + 0.447i
Analytic conductor: 32.339432.3394
Root analytic conductor: 5.686775.68677
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4050(649,⋅)\chi_{4050} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4050, ( :1/2), −0.894+0.447i)(2,\ 4050,\ (\ :1/2),\ -0.894 + 0.447i)

Particular Values

L(1)L(1) ≈\approx 0.95127367480.9512736748
L(12)L(\frac12) ≈\approx 0.95127367480.9512736748
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 1−4iT−7T2 1 - 4iT - 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1+iT−13T2 1 + iT - 13T^{2}
17 1−3iT−17T2 1 - 3iT - 17T^{2}
19 1−4T+19T2 1 - 4T + 19T^{2}
23 1−23T2 1 - 23T^{2}
29 1+9T+29T2 1 + 9T + 29T^{2}
31 1+4T+31T2 1 + 4T + 31T^{2}
37 1−iT−37T2 1 - iT - 37T^{2}
41 1−6T+41T2 1 - 6T + 41T^{2}
43 1−8iT−43T2 1 - 8iT - 43T^{2}
47 1−12iT−47T2 1 - 12iT - 47T^{2}
53 1+6iT−53T2 1 + 6iT - 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1+T+61T2 1 + T + 61T^{2}
67 1−4iT−67T2 1 - 4iT - 67T^{2}
71 1+12T+71T2 1 + 12T + 71T^{2}
73 1−11iT−73T2 1 - 11iT - 73T^{2}
79 1−16T+79T2 1 - 16T + 79T^{2}
83 1+12iT−83T2 1 + 12iT - 83T^{2}
89 1−3T+89T2 1 - 3T + 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

−8.900544546637272776672958934026, −7.993936650441777623983882154210, −7.56364463833830175531353236062, −6.53613904393485464906030937297, −5.74879838736197856199114861138, −5.49799113904488327787955516363, −4.50304757635060855614619496632, −3.49041304798277421477860532803, −2.60607115193411289020490299936, −1.48858910180921083050740063540, 0.28430474715581416215483635426, 1.29328879913571905920769689735, 2.34938286126865355307774196393, 3.57067866155828413075964481022, 3.89034624218382587703627909221, 4.89942545675181924895369893307, 5.57894157718129079984176208177, 6.73582501922805070921745560488, 7.43440074110954368404356386468, 7.81900891248323063813513522436

Graph of the ZZ-function along the critical line