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Properties

Label 2-1305-145.144-c1-0-60
Degree 22
Conductor 13051305
Sign 0.0830+0.996i0.0830 + 0.996i
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  + 2·2-s + 2·4-s + (−2 + i)5-s − 4i·7-s + (−4 + 2i)10-s + i·11-s − 2i·13-s − 8i·14-s − 4·16-s + 6·17-s − 4i·19-s + (−4 + 2i)20-s + 2i·22-s − 9i·23-s + (3 − 4i)25-s − 4i·26-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + (−0.894 + 0.447i)5-s − 1.51i·7-s + (−1.26 + 0.632i)10-s + 0.301i·11-s − 0.554i·13-s − 2.13i·14-s − 16-s + 1.45·17-s − 0.917i·19-s + (−0.894 + 0.447i)20-s + 0.426i·22-s − 1.87i·23-s + (0.600 − 0.800i)25-s − 0.784i·26-s + ⋯

Functional equation

Λ(s)=(1305s/2 ΓC(s) L(s)=((0.0830+0.996i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0830 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1305s/2 ΓC(s+1/2) L(s)=((0.0830+0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0830 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13051305    =    32⋅5⋅293^{2} \cdot 5 \cdot 29
Sign: 0.0830+0.996i0.0830 + 0.996i
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), 0.0830+0.996i)(2,\ 1305,\ (\ :1/2),\ 0.0830 + 0.996i)

Particular Values

L(1)L(1) ≈\approx 2.4534819132.453481913
L(12)L(\frac12) ≈\approx 2.4534819132.453481913
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+(2−i)T 1 + (2 - i)T
29 1+(−2+5i)T 1 + (-2 + 5i)T
good2 1−2T+2T2 1 - 2T + 2T^{2}
7 1+4iT−7T2 1 + 4iT - 7T^{2}
11 1−iT−11T2 1 - iT - 11T^{2}
13 1+2iT−13T2 1 + 2iT - 13T^{2}
17 1−6T+17T2 1 - 6T + 17T^{2}
19 1+4iT−19T2 1 + 4iT - 19T^{2}
23 1+9iT−23T2 1 + 9iT - 23T^{2}
31 1+2iT−31T2 1 + 2iT - 31T^{2}
37 1+T+37T2 1 + T + 37T^{2}
41 1−9iT−41T2 1 - 9iT - 41T^{2}
43 1+T+43T2 1 + T + 43T^{2}
47 1+8T+47T2 1 + 8T + 47T^{2}
53 1−9iT−53T2 1 - 9iT - 53T^{2}
59 1+8T+59T2 1 + 8T + 59T^{2}
61 1+6iT−61T2 1 + 6iT - 61T^{2}
67 1−12iT−67T2 1 - 12iT - 67T^{2}
71 1+2T+71T2 1 + 2T + 71T^{2}
73 1−15T+73T2 1 - 15T + 73T^{2}
79 1−4iT−79T2 1 - 4iT - 79T^{2}
83 1+7iT−83T2 1 + 7iT - 83T^{2}
89 1+2iT−89T2 1 + 2iT - 89T^{2}
97 1−11T+97T2 1 - 11T + 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

−9.738429152327209098155837035283, −8.302843257897433570293909223490, −7.60973984109287156553467338298, −6.83579944325227858511132406863, −6.14885502234064744496038314249, −4.80012059859727431091523076309, −4.37797694144427953366951522619, −3.47663691456538662284798495539, −2.76150253355720034799778383644, −0.64223607711012492924852643119, 1.77088019764797899931344369182, 3.28478753548417363019912845808, 3.59254007917073859749384693798, 4.95488288955330625006751518500, 5.40613569696574780957360186736, 6.13340633299569553057515089107, 7.27300529849285495151592315098, 8.207370617276444456773597116117, 8.950893201578666372078248978392, 9.705241165562487214321846287697

Graph of the ZZ-function along the critical line