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Properties

Label 2-1472-1472.965-c0-0-0
Degree 22
Conductor 14721472
Sign 0.956+0.290i0.956 + 0.290i
Analytic cond. 0.7346230.734623
Root an. cond. 0.8571010.857101
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.130 + 0.991i)2-s + (−1.65 − 1.10i)3-s + (−0.965 − 0.258i)4-s + (1.31 − 1.50i)6-s + (0.382 − 0.923i)8-s + (1.14 + 2.75i)9-s + (1.31 + 1.50i)12-s + (−0.867 + 0.172i)13-s + (0.866 + 0.5i)16-s + (−2.88 + 0.772i)18-s + (−0.923 + 0.382i)23-s + (−1.65 + 1.10i)24-s + (0.923 + 0.382i)25-s + (−0.0578 − 0.882i)26-s + (0.772 − 3.88i)27-s + ⋯
L(s)  = 1  + (−0.130 + 0.991i)2-s + (−1.65 − 1.10i)3-s + (−0.965 − 0.258i)4-s + (1.31 − 1.50i)6-s + (0.382 − 0.923i)8-s + (1.14 + 2.75i)9-s + (1.31 + 1.50i)12-s + (−0.867 + 0.172i)13-s + (0.866 + 0.5i)16-s + (−2.88 + 0.772i)18-s + (−0.923 + 0.382i)23-s + (−1.65 + 1.10i)24-s + (0.923 + 0.382i)25-s + (−0.0578 − 0.882i)26-s + (0.772 − 3.88i)27-s + ⋯

Functional equation

Λ(s)=(1472s/2 ΓC(s) L(s)=((0.956+0.290i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1472s/2 ΓC(s) L(s)=((0.956+0.290i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14721472    =    26⋅232^{6} \cdot 23
Sign: 0.956+0.290i0.956 + 0.290i
Analytic conductor: 0.7346230.734623
Root analytic conductor: 0.8571010.857101
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1472(965,⋅)\chi_{1472} (965, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1472, ( :0), 0.956+0.290i)(2,\ 1472,\ (\ :0),\ 0.956 + 0.290i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.45438292120.4543829212
L(12)L(\frac12) ≈\approx 0.45438292120.4543829212
L(1)L(1) 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+(0.130−0.991i)T 1 + (0.130 - 0.991i)T
23 1+(0.923−0.382i)T 1 + (0.923 - 0.382i)T
good3 1+(1.65+1.10i)T+(0.382+0.923i)T2 1 + (1.65 + 1.10i)T + (0.382 + 0.923i)T^{2}
5 1+(−0.923−0.382i)T2 1 + (-0.923 - 0.382i)T^{2}
7 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
11 1+(0.382−0.923i)T2 1 + (0.382 - 0.923i)T^{2}
13 1+(0.867−0.172i)T+(0.923−0.382i)T2 1 + (0.867 - 0.172i)T + (0.923 - 0.382i)T^{2}
17 1+iT2 1 + iT^{2}
19 1+(0.923−0.382i)T2 1 + (0.923 - 0.382i)T^{2}
29 1+(−0.835+1.25i)T+(−0.382−0.923i)T2 1 + (-0.835 + 1.25i)T + (-0.382 - 0.923i)T^{2}
31 1+0.261iT−T2 1 + 0.261iT - T^{2}
37 1+(0.923+0.382i)T2 1 + (0.923 + 0.382i)T^{2}
41 1+(−0.923+0.382i)T+(0.707−0.707i)T2 1 + (-0.923 + 0.382i)T + (0.707 - 0.707i)T^{2}
43 1+(−0.382+0.923i)T2 1 + (-0.382 + 0.923i)T^{2}
47 1+(−1.36+1.36i)T−iT2 1 + (-1.36 + 1.36i)T - iT^{2}
53 1+(0.382−0.923i)T2 1 + (0.382 - 0.923i)T^{2}
59 1+(−1.63−0.324i)T+(0.923+0.382i)T2 1 + (-1.63 - 0.324i)T + (0.923 + 0.382i)T^{2}
61 1+(−0.382−0.923i)T2 1 + (-0.382 - 0.923i)T^{2}
67 1+(−0.382−0.923i)T2 1 + (-0.382 - 0.923i)T^{2}
71 1+(−0.465+1.12i)T+(−0.707−0.707i)T2 1 + (-0.465 + 1.12i)T + (-0.707 - 0.707i)T^{2}
73 1+(0.198+0.478i)T+(−0.707+0.707i)T2 1 + (0.198 + 0.478i)T + (-0.707 + 0.707i)T^{2}
79 1−iT2 1 - iT^{2}
83 1+(0.923−0.382i)T2 1 + (0.923 - 0.382i)T^{2}
89 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
97 1+T2 1 + T^{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

−9.811772040876310979630652854476, −8.521398670136778805968654425278, −7.65950847409518756285824197742, −7.15075675632878625126705799754, −6.42713481063632636894959209169, −5.73138629060358143600060927331, −5.05782910978178391219082631047, −4.23820124001746296333588215319, −2.10730566676595812993243881948, −0.65235050718403857882520871094, 0.978783891593516166102316244008, 2.81704852263782768181217729860, 3.97380501435881725004539130005, 4.66669424761420122470079992938, 5.28147314242715698096527814604, 6.19246428182255158183765557650, 7.18559625719388141021227405883, 8.505959209877949006634363072365, 9.416305322388503758347332641281, 9.951497916503560686363752477532

Graph of the ZZ-function along the critical line