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Properties

Label 2-1305-29.28-c1-0-28
Degree 22
Conductor 13051305
Sign 0.964+0.262i0.964 + 0.262i
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  − 0.517i·2-s + 1.73·4-s + 5-s + 0.732·7-s − 1.93i·8-s − 0.517i·10-s + 5.27i·11-s − 1.46·13-s − 0.378i·14-s + 2.46·16-s − 6.31i·17-s + 4.24i·19-s + 1.73·20-s + 2.73·22-s + 8.19·23-s + ⋯
L(s)  = 1  − 0.366i·2-s + 0.866·4-s + 0.447·5-s + 0.276·7-s − 0.683i·8-s − 0.163i·10-s + 1.59i·11-s − 0.406·13-s − 0.101i·14-s + 0.616·16-s − 1.53i·17-s + 0.973i·19-s + 0.387·20-s + 0.582·22-s + 1.70·23-s + ⋯

Functional equation

Λ(s)=(1305s/2 ΓC(s) L(s)=((0.964+0.262i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1305s/2 ΓC(s+1/2) L(s)=((0.964+0.262i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13051305    =    32⋅5⋅293^{2} \cdot 5 \cdot 29
Sign: 0.964+0.262i0.964 + 0.262i
Analytic conductor: 10.420410.4204
Root analytic conductor: 3.228073.22807
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1305(811,⋅)\chi_{1305} (811, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1305, ( :1/2), 0.964+0.262i)(2,\ 1305,\ (\ :1/2),\ 0.964 + 0.262i)

Particular Values

L(1)L(1) ≈\approx 2.3958375202.395837520
L(12)L(\frac12) ≈\approx 2.3958375202.395837520
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−T 1 - T
29 1+(−5.19−1.41i)T 1 + (-5.19 - 1.41i)T
good2 1+0.517iT−2T2 1 + 0.517iT - 2T^{2}
7 1−0.732T+7T2 1 - 0.732T + 7T^{2}
11 1−5.27iT−11T2 1 - 5.27iT - 11T^{2}
13 1+1.46T+13T2 1 + 1.46T + 13T^{2}
17 1+6.31iT−17T2 1 + 6.31iT - 17T^{2}
19 1−4.24iT−19T2 1 - 4.24iT - 19T^{2}
23 1−8.19T+23T2 1 - 8.19T + 23T^{2}
31 1−4.24iT−31T2 1 - 4.24iT - 31T^{2}
37 1+4.24iT−37T2 1 + 4.24iT - 37T^{2}
41 1+8.76iT−41T2 1 + 8.76iT - 41T^{2}
43 1−4.24iT−43T2 1 - 4.24iT - 43T^{2}
47 1−8.38iT−47T2 1 - 8.38iT - 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1+6T+59T2 1 + 6T + 59T^{2}
61 1+3.10iT−61T2 1 + 3.10iT - 61T^{2}
67 1−11.1T+67T2 1 - 11.1T + 67T^{2}
71 1+6T+71T2 1 + 6T + 71T^{2}
73 1−1.13iT−73T2 1 - 1.13iT - 73T^{2}
79 1+15.8iT−79T2 1 + 15.8iT - 79T^{2}
83 1+2.19T+83T2 1 + 2.19T + 83T^{2}
89 1+2.07iT−89T2 1 + 2.07iT - 89T^{2}
97 1−7.34iT−97T2 1 - 7.34iT - 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

−9.707458107138268132647946684346, −9.090247201310059505531594213825, −7.72037653526070772012876409647, −7.15837899811132855352686026312, −6.51524986976371869489971150388, −5.27082211197452794400160972628, −4.60423143111177424650162524335, −3.16419386654685291055523922478, −2.32544833326335750309346553924, −1.32343538920174694371640696452, 1.19059846704778186101725998641, 2.51888163951511472130820667079, 3.34477138336643196267959678122, 4.80612519147083750021391927206, 5.69670812472436997251163372644, 6.38731141026002586145336221525, 7.05399038328580424735854791316, 8.243335259873889515227940987087, 8.530723771379059900775758159503, 9.696792571180074873440301512654

Graph of the ZZ-function along the critical line