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Properties

Label 2-1305-5.4-c1-0-50
Degree 22
Conductor 13051305
Sign −0.447+0.894i-0.447 + 0.894i
Analytic cond. 10.420410.4204
Root an. cond. 3.228073.22807
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 + (−1 + 2i)5-s − 2i·7-s − 3i·8-s + (2 + i)10-s − 4i·13-s − 2·14-s − 16-s − 2i·17-s + (−1 + 2i)20-s + 2i·23-s + (−3 − 4i)25-s − 4·26-s − 2i·28-s + 29-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.5·4-s + (−0.447 + 0.894i)5-s − 0.755i·7-s − 1.06i·8-s + (0.632 + 0.316i)10-s − 1.10i·13-s − 0.534·14-s − 0.250·16-s − 0.485i·17-s + (−0.223 + 0.447i)20-s + 0.417i·23-s + (−0.600 − 0.800i)25-s − 0.784·26-s − 0.377i·28-s + 0.185·29-s + ⋯

Functional equation

Λ(s)=(1305s/2 ΓC(s) L(s)=((−0.447+0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1305s/2 ΓC(s+1/2) L(s)=((−0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{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: 13051305    =    32⋅5⋅293^{2} \cdot 5 \cdot 29
Sign: −0.447+0.894i-0.447 + 0.894i
Analytic conductor: 10.420410.4204
Root analytic conductor: 3.228073.22807
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1305(784,⋅)\chi_{1305} (784, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1305, ( :1/2), −0.447+0.894i)(2,\ 1305,\ (\ :1/2),\ -0.447 + 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.5925196151.592519615
L(12)L(\frac12) ≈\approx 1.5925196151.592519615
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 1
5 1+(1−2i)T 1 + (1 - 2i)T
29 1−T 1 - T
good2 1+iT−2T2 1 + iT - 2T^{2}
7 1+2iT−7T2 1 + 2iT - 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1+4iT−13T2 1 + 4iT - 13T^{2}
17 1+2iT−17T2 1 + 2iT - 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1−2iT−23T2 1 - 2iT - 23T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1−2iT−37T2 1 - 2iT - 37T^{2}
41 1+10T+41T2 1 + 10T + 41T^{2}
43 1−43T2 1 - 43T^{2}
47 1+12iT−47T2 1 + 12iT - 47T^{2}
53 1+12iT−53T2 1 + 12iT - 53T^{2}
59 1−4T+59T2 1 - 4T + 59T^{2}
61 1−2T+61T2 1 - 2T + 61T^{2}
67 1−2iT−67T2 1 - 2iT - 67T^{2}
71 1−8T+71T2 1 - 8T + 71T^{2}
73 1+14iT−73T2 1 + 14iT - 73T^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1+6iT−83T2 1 + 6iT - 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1−10iT−97T2 1 - 10iT - 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

−10.00505846949518651350295593141, −8.485499092341623796456950824390, −7.60242132995005891459753887582, −7.00451176912474970652342581349, −6.28641752581960476903018400151, −5.03942382321404709281914026419, −3.71728796090219176312261600033, −3.23711203771078452186518527739, −2.17323644387502277931074269001, −0.65504233145492505097997711043, 1.55265502428620737703863827011, 2.67440266786593693114220106013, 4.10809068241645342297028270241, 4.98732694313822228938769051418, 5.84487695249309432141090844679, 6.59130344346890757875140372092, 7.46510970093759013940925842897, 8.358543160738460753766710413005, 8.773355966290434546116607870882, 9.673208401878116675694158808760

Graph of the ZZ-function along the critical line