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Properties

Label 2-1305-145.144-c1-0-0
Degree 22
Conductor 13051305
Sign −0.991−0.126i-0.991 - 0.126i
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  − 2.23·2-s + 3.00·4-s + (1 + 2i)5-s + 2i·7-s − 2.23·8-s + (−2.23 − 4.47i)10-s − 4.47i·11-s − 4i·13-s − 4.47i·14-s − 0.999·16-s − 4.47·17-s + 4.47i·19-s + (3.00 + 6.00i)20-s + 10.0i·22-s + 6i·23-s + ⋯
L(s)  = 1  − 1.58·2-s + 1.50·4-s + (0.447 + 0.894i)5-s + 0.755i·7-s − 0.790·8-s + (−0.707 − 1.41i)10-s − 1.34i·11-s − 1.10i·13-s − 1.19i·14-s − 0.249·16-s − 1.08·17-s + 1.02i·19-s + (0.670 + 1.34i)20-s + 2.13i·22-s + 1.25i·23-s + ⋯

Functional equation

Λ(s)=(1305s/2 ΓC(s) L(s)=((−0.991−0.126i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1305s/2 ΓC(s+1/2) L(s)=((−0.991−0.126i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13051305    =    32⋅5⋅293^{2} \cdot 5 \cdot 29
Sign: −0.991−0.126i-0.991 - 0.126i
Analytic conductor: 10.420410.4204
Root analytic conductor: 3.228073.22807
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1305(289,⋅)\chi_{1305} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1305, ( :1/2), −0.991−0.126i)(2,\ 1305,\ (\ :1/2),\ -0.991 - 0.126i)

Particular Values

L(1)L(1) ≈\approx 0.23527698560.2352769856
L(12)L(\frac12) ≈\approx 0.23527698560.2352769856
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+(3−4.47i)T 1 + (3 - 4.47i)T
good2 1+2.23T+2T2 1 + 2.23T + 2T^{2}
7 1−2iT−7T2 1 - 2iT - 7T^{2}
11 1+4.47iT−11T2 1 + 4.47iT - 11T^{2}
13 1+4iT−13T2 1 + 4iT - 13T^{2}
17 1+4.47T+17T2 1 + 4.47T + 17T^{2}
19 1−4.47iT−19T2 1 - 4.47iT - 19T^{2}
23 1−6iT−23T2 1 - 6iT - 23T^{2}
31 1+4.47iT−31T2 1 + 4.47iT - 31T^{2}
37 1+4.47T+37T2 1 + 4.47T + 37T^{2}
41 1−41T2 1 - 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1+8.94T+47T2 1 + 8.94T + 47T^{2}
53 1+4iT−53T2 1 + 4iT - 53T^{2}
59 1+4T+59T2 1 + 4T + 59T^{2}
61 1−8.94iT−61T2 1 - 8.94iT - 61T^{2}
67 1+2iT−67T2 1 + 2iT - 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+13.4T+73T2 1 + 13.4T + 73T^{2}
79 1−13.4iT−79T2 1 - 13.4iT - 79T^{2}
83 1+6iT−83T2 1 + 6iT - 83T^{2}
89 1−17.8iT−89T2 1 - 17.8iT - 89T^{2}
97 1−4.47T+97T2 1 - 4.47T + 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

−9.935738535811924421416301655404, −9.210825985635991103754745299279, −8.499357806274984232468047488467, −7.84748978019647332716369706895, −6.98626991785551425940130702943, −6.04387031888616029287739630217, −5.47331098420434753368379248904, −3.54740027338595141059816997852, −2.63801502454215859539058414259, −1.54946195645694292037376711820, 0.16288221280421833640861376902, 1.56097472786110788727540770295, 2.29797299929811916202021264823, 4.37214958392142835310739236006, 4.74544196587878280584759612203, 6.46890242925906042264335539151, 6.95726196215900363794146718354, 7.76450723859041965701026358941, 8.819287887331014677195795881103, 9.096349953110822791115792883863

Graph of the ZZ-function along the critical line