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Properties

Label 2-3120-5.4-c1-0-45
Degree 22
Conductor 31203120
Sign −0.683+0.730i-0.683 + 0.730i
Analytic cond. 24.913324.9133
Root an. cond. 4.991324.99132
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·3-s + (−1.52 + 1.63i)5-s + 2.82i·7-s − 9-s − 1.05·11-s − i·13-s + (1.63 + 1.52i)15-s + 7.14i·17-s − 6.61·19-s + 2.82·21-s − 3.49i·23-s + (−0.332 − 4.98i)25-s + i·27-s − 4.82·29-s + 9.65·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.683 + 0.730i)5-s + 1.06i·7-s − 0.333·9-s − 0.318·11-s − 0.277i·13-s + (0.421 + 0.394i)15-s + 1.73i·17-s − 1.51·19-s + 0.617·21-s − 0.728i·23-s + (−0.0664 − 0.997i)25-s + 0.192i·27-s − 0.896·29-s + 1.73·31-s + ⋯

Functional equation

Λ(s)=(3120s/2 ΓC(s) L(s)=((−0.683+0.730i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3120s/2 ΓC(s+1/2) L(s)=((−0.683+0.730i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31203120    =    24⋅3⋅5⋅132^{4} \cdot 3 \cdot 5 \cdot 13
Sign: −0.683+0.730i-0.683 + 0.730i
Analytic conductor: 24.913324.9133
Root analytic conductor: 4.991324.99132
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3120(1249,⋅)\chi_{3120} (1249, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3120, ( :1/2), −0.683+0.730i)(2,\ 3120,\ (\ :1/2),\ -0.683 + 0.730i)

Particular Values

L(1)L(1) ≈\approx 0.19488512700.1948851270
L(12)L(\frac12) ≈\approx 0.19488512700.1948851270
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+iT 1 + iT
5 1+(1.52−1.63i)T 1 + (1.52 - 1.63i)T
13 1+iT 1 + iT
good7 1−2.82iT−7T2 1 - 2.82iT - 7T^{2}
11 1+1.05T+11T2 1 + 1.05T + 11T^{2}
17 1−7.14iT−17T2 1 - 7.14iT - 17T^{2}
19 1+6.61T+19T2 1 + 6.61T + 19T^{2}
23 1+3.49iT−23T2 1 + 3.49iT - 23T^{2}
29 1+4.82T+29T2 1 + 4.82T + 29T^{2}
31 1−9.65T+31T2 1 - 9.65T + 31T^{2}
37 1−8.11iT−37T2 1 - 8.11iT - 37T^{2}
41 1+3.26T+41T2 1 + 3.26T + 41T^{2}
43 1+7.44iT−43T2 1 + 7.44iT - 43T^{2}
47 1+11.3iT−47T2 1 + 11.3iT - 47T^{2}
53 1+4.53iT−53T2 1 + 4.53iT - 53T^{2}
59 1−1.05T+59T2 1 - 1.05T + 59T^{2}
61 1+7.97T+61T2 1 + 7.97T + 61T^{2}
67 1+1.46iT−67T2 1 + 1.46iT - 67T^{2}
71 1+0.845T+71T2 1 + 0.845T + 71T^{2}
73 1+9.28iT−73T2 1 + 9.28iT - 73T^{2}
79 1−6.85T+79T2 1 - 6.85T + 79T^{2}
83 1−7.97iT−83T2 1 - 7.97iT - 83T^{2}
89 1+0.139T+89T2 1 + 0.139T + 89T^{2}
97 1+6.93iT−97T2 1 + 6.93iT - 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

−8.402407361315103793124230735030, −7.85311718455695404659388256840, −6.69035395019837956672926682602, −6.37501365657182000387913973884, −5.54088866960617883482889753005, −4.45172146515037995679086453012, −3.54343448025242013807414623992, −2.58896335430893605521109856622, −1.87809108011991915347181844556, −0.06676320228470484738520541032, 1.06623437779932593396757941156, 2.59083435534561477255637643749, 3.64672352181339885601468195452, 4.42632356399380836381523050340, 4.78786738971404626286941693017, 5.83083046546918505706731291634, 6.86215367370877749295511865189, 7.59874935111427237945299584349, 8.131590339311971337344660529255, 9.134697437031830259140441140198

Graph of the ZZ-function along the critical line