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Properties

Label 2-1472-16.13-c1-0-40
Degree 22
Conductor 14721472
Sign −0.694+0.719i-0.694 + 0.719i
Analytic cond. 11.753911.7539
Root an. cond. 3.428403.42840
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  + (−0.339 − 0.339i)3-s + (1.46 − 1.46i)5-s − 2.63i·7-s − 2.76i·9-s + (2.99 − 2.99i)11-s + (−0.749 − 0.749i)13-s − 0.998·15-s − 3.64·17-s + (−1.76 − 1.76i)19-s + (−0.893 + 0.893i)21-s + i·23-s + 0.682i·25-s + (−1.95 + 1.95i)27-s + (0.0790 + 0.0790i)29-s + 2.07·31-s + ⋯
L(s)  = 1  + (−0.196 − 0.196i)3-s + (0.657 − 0.657i)5-s − 0.994i·7-s − 0.923i·9-s + (0.901 − 0.901i)11-s + (−0.207 − 0.207i)13-s − 0.257·15-s − 0.885·17-s + (−0.405 − 0.405i)19-s + (−0.195 + 0.195i)21-s + 0.208i·23-s + 0.136i·25-s + (−0.377 + 0.377i)27-s + (0.0146 + 0.0146i)29-s + 0.372·31-s + ⋯

Functional equation

Λ(s)=(1472s/2 ΓC(s) L(s)=((−0.694+0.719i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1472s/2 ΓC(s+1/2) L(s)=((−0.694+0.719i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14721472    =    26⋅232^{6} \cdot 23
Sign: −0.694+0.719i-0.694 + 0.719i
Analytic conductor: 11.753911.7539
Root analytic conductor: 3.428403.42840
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1472(1105,⋅)\chi_{1472} (1105, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1472, ( :1/2), −0.694+0.719i)(2,\ 1472,\ (\ :1/2),\ -0.694 + 0.719i)

Particular Values

L(1)L(1) ≈\approx 1.5314847301.531484730
L(12)L(\frac12) ≈\approx 1.5314847301.531484730
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)
bad2 1 1
23 1−iT 1 - iT
good3 1+(0.339+0.339i)T+3iT2 1 + (0.339 + 0.339i)T + 3iT^{2}
5 1+(−1.46+1.46i)T−5iT2 1 + (-1.46 + 1.46i)T - 5iT^{2}
7 1+2.63iT−7T2 1 + 2.63iT - 7T^{2}
11 1+(−2.99+2.99i)T−11iT2 1 + (-2.99 + 2.99i)T - 11iT^{2}
13 1+(0.749+0.749i)T+13iT2 1 + (0.749 + 0.749i)T + 13iT^{2}
17 1+3.64T+17T2 1 + 3.64T + 17T^{2}
19 1+(1.76+1.76i)T+19iT2 1 + (1.76 + 1.76i)T + 19iT^{2}
29 1+(−0.0790−0.0790i)T+29iT2 1 + (-0.0790 - 0.0790i)T + 29iT^{2}
31 1−2.07T+31T2 1 - 2.07T + 31T^{2}
37 1+(3.64−3.64i)T−37iT2 1 + (3.64 - 3.64i)T - 37iT^{2}
41 1−1.90iT−41T2 1 - 1.90iT - 41T^{2}
43 1+(6.84−6.84i)T−43iT2 1 + (6.84 - 6.84i)T - 43iT^{2}
47 1−8.55T+47T2 1 - 8.55T + 47T^{2}
53 1+(6.62−6.62i)T−53iT2 1 + (6.62 - 6.62i)T - 53iT^{2}
59 1+(−5.45+5.45i)T−59iT2 1 + (-5.45 + 5.45i)T - 59iT^{2}
61 1+(−2.13−2.13i)T+61iT2 1 + (-2.13 - 2.13i)T + 61iT^{2}
67 1+(3.51+3.51i)T+67iT2 1 + (3.51 + 3.51i)T + 67iT^{2}
71 1−1.61iT−71T2 1 - 1.61iT - 71T^{2}
73 1+14.4iT−73T2 1 + 14.4iT - 73T^{2}
79 1−12.5T+79T2 1 - 12.5T + 79T^{2}
83 1+(1.26+1.26i)T+83iT2 1 + (1.26 + 1.26i)T + 83iT^{2}
89 1+12.3iT−89T2 1 + 12.3iT - 89T^{2}
97 1−7.54T+97T2 1 - 7.54T + 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

−9.136048999898600361553651747593, −8.647850105369341503831675214073, −7.50664723562352460004876338054, −6.56152091720335596977640825455, −6.14981092234334337336735138871, −5.00248054576824184220464406610, −4.11158516940496739591226015037, −3.18166786802066162672721783911, −1.57388281266249956243126653540, −0.62348149970286958057789895013, 2.00972565236967846728183443209, 2.39066664833101758979989193297, 3.93138770126395658925667866546, 4.87252417226032992611450445880, 5.71185550177479787483435901757, 6.55182171454828723040564485163, 7.17726576265476309452070821562, 8.373995017107906688388914553260, 9.034749965875839601890038462067, 9.901006615499911310939795627857

Graph of the ZZ-function along the critical line