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Properties

Label 2-1305-145.144-c1-0-47
Degree 22
Conductor 13051305
Sign 0.747+0.664i0.747 + 0.664i
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-s − 4-s + (1 + 2i)5-s − 2i·7-s − 3·8-s + (1 + 2i)10-s − 2i·11-s − 4i·13-s − 2i·14-s − 16-s + 6·17-s + 2i·19-s + (−1 − 2i)20-s − 2i·22-s − 6i·23-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.5·4-s + (0.447 + 0.894i)5-s − 0.755i·7-s − 1.06·8-s + (0.316 + 0.632i)10-s − 0.603i·11-s − 1.10i·13-s − 0.534i·14-s − 0.250·16-s + 1.45·17-s + 0.458i·19-s + (−0.223 − 0.447i)20-s − 0.426i·22-s − 1.25i·23-s + ⋯

Functional equation

Λ(s)=(1305s/2 ΓC(s) L(s)=((0.747+0.664i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1305s/2 ΓC(s+1/2) L(s)=((0.747+0.664i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13051305    =    32⋅5⋅293^{2} \cdot 5 \cdot 29
Sign: 0.747+0.664i0.747 + 0.664i
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), 0.747+0.664i)(2,\ 1305,\ (\ :1/2),\ 0.747 + 0.664i)

Particular Values

L(1)L(1) ≈\approx 2.0099734082.009973408
L(12)L(\frac12) ≈\approx 2.0099734082.009973408
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+(−5+2i)T 1 + (-5 + 2i)T
good2 1−T+2T2 1 - T + 2T^{2}
7 1+2iT−7T2 1 + 2iT - 7T^{2}
11 1+2iT−11T2 1 + 2iT - 11T^{2}
13 1+4iT−13T2 1 + 4iT - 13T^{2}
17 1−6T+17T2 1 - 6T + 17T^{2}
19 1−2iT−19T2 1 - 2iT - 19T^{2}
23 1+6iT−23T2 1 + 6iT - 23T^{2}
31 1+2iT−31T2 1 + 2iT - 31T^{2}
37 1+2T+37T2 1 + 2T + 37T^{2}
41 1−41T2 1 - 41T^{2}
43 1−4T+43T2 1 - 4T + 43T^{2}
47 1−8T+47T2 1 - 8T + 47T^{2}
53 1+12iT−53T2 1 + 12iT - 53T^{2}
59 1−4T+59T2 1 - 4T + 59T^{2}
61 1−12iT−61T2 1 - 12iT - 61T^{2}
67 1+6iT−67T2 1 + 6iT - 67T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1+6T+73T2 1 + 6T + 73T^{2}
79 1+2iT−79T2 1 + 2iT - 79T^{2}
83 1+2iT−83T2 1 + 2iT - 83T^{2}
89 1−16iT−89T2 1 - 16iT - 89T^{2}
97 1+14T+97T2 1 + 14T + 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

−9.902337301472496733139408104652, −8.646963565966983029106201930584, −7.913312280305588642713215425180, −6.97665694776657729129051438115, −5.92760230442639695388624479270, −5.53423337621511941801479793110, −4.29625288946228132058118414308, −3.43042467485259959085456193143, −2.72835092344031485649477893492, −0.76393714206530891578627559963, 1.32689956029820022363625464688, 2.66129910792525779657171666232, 3.88548757733506825570768597457, 4.74299492762490571545970974386, 5.42650341598614599145451380584, 6.03944081049483700045557467413, 7.22676304236838813091087026639, 8.311602364099174959841546770019, 9.166898453577486070199159032513, 9.402095747839409739074834278817

Graph of the ZZ-function along the critical line