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Properties

Label 2-1014-13.12-c3-0-61
Degree 22
Conductor 10141014
Sign −0.554+0.832i-0.554 + 0.832i
Analytic cond. 59.827959.8279
Root an. cond. 7.734857.73485
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s + 3·3-s − 4·4-s + 10i·5-s − 6i·6-s + 8i·7-s + 8i·8-s + 9·9-s + 20·10-s − 40i·11-s − 12·12-s + 16·14-s + 30i·15-s + 16·16-s − 130·17-s − 18i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.894i·5-s − 0.408i·6-s + 0.431i·7-s + 0.353i·8-s + 0.333·9-s + 0.632·10-s − 1.09i·11-s − 0.288·12-s + 0.305·14-s + 0.516i·15-s + 0.250·16-s − 1.85·17-s − 0.235i·18-s + ⋯

Functional equation

Λ(s)=(1014s/2 ΓC(s) L(s)=((−0.554+0.832i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1014s/2 ΓC(s+3/2) L(s)=((−0.554+0.832i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10141014    =    2⋅3⋅1322 \cdot 3 \cdot 13^{2}
Sign: −0.554+0.832i-0.554 + 0.832i
Analytic conductor: 59.827959.8279
Root analytic conductor: 7.734857.73485
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1014(337,⋅)\chi_{1014} (337, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1014, ( :3/2), −0.554+0.832i)(2,\ 1014,\ (\ :3/2),\ -0.554 + 0.832i)

Particular Values

L(2)L(2) ≈\approx 1.4823927711.482392771
L(12)L(\frac12) ≈\approx 1.4823927711.482392771
L(52)L(\frac{5}{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)
bad2 1+2iT 1 + 2iT
3 1−3T 1 - 3T
13 1 1
good5 1−10iT−125T2 1 - 10iT - 125T^{2}
7 1−8iT−343T2 1 - 8iT - 343T^{2}
11 1+40iT−1.33e3T2 1 + 40iT - 1.33e3T^{2}
17 1+130T+4.91e3T2 1 + 130T + 4.91e3T^{2}
19 1+20iT−6.85e3T2 1 + 20iT - 6.85e3T^{2}
23 1+1.21e4T2 1 + 1.21e4T^{2}
29 1+18T+2.43e4T2 1 + 18T + 2.43e4T^{2}
31 1+184iT−2.97e4T2 1 + 184iT - 2.97e4T^{2}
37 1−74iT−5.06e4T2 1 - 74iT - 5.06e4T^{2}
41 1+362iT−6.89e4T2 1 + 362iT - 6.89e4T^{2}
43 1+76T+7.95e4T2 1 + 76T + 7.95e4T^{2}
47 1−452iT−1.03e5T2 1 - 452iT - 1.03e5T^{2}
53 1−382T+1.48e5T2 1 - 382T + 1.48e5T^{2}
59 1+464iT−2.05e5T2 1 + 464iT - 2.05e5T^{2}
61 1−358T+2.26e5T2 1 - 358T + 2.26e5T^{2}
67 1+700iT−3.00e5T2 1 + 700iT - 3.00e5T^{2}
71 1+748iT−3.57e5T2 1 + 748iT - 3.57e5T^{2}
73 1+1.05e3iT−3.89e5T2 1 + 1.05e3iT - 3.89e5T^{2}
79 1+976T+4.93e5T2 1 + 976T + 4.93e5T^{2}
83 1+1.00e3iT−5.71e5T2 1 + 1.00e3iT - 5.71e5T^{2}
89 1−386iT−7.04e5T2 1 - 386iT - 7.04e5T^{2}
97 1+614iT−9.12e5T2 1 + 614iT - 9.12e5T^{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

−9.124794221872116073003463240844, −8.789616090104045527964856577049, −7.76612767480301943300680066515, −6.73999239465243409347204606476, −5.92104777030879332334984403212, −4.64622862781921420867562534243, −3.62735962932438679534243527039, −2.76813292916274978802455244177, −2.03829401380890338725458088501, −0.35420853440194243953923057116, 1.16923579954615080831175118171, 2.41494561031306130401015048528, 4.01603530283211187039009158866, 4.55362670802625689011220061591, 5.45676042879623974439533014159, 6.85062451753392867027536244855, 7.16165388409966698832172935135, 8.458817569423969188568435150628, 8.706375114149408590148870097491, 9.670128744536759041032034745284

Graph of the ZZ-function along the critical line