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Properties

Label 2-1472-368.45-c0-0-2
Degree 22
Conductor 14721472
Sign −0.923+0.382i-0.923 + 0.382i
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  + (−1 − i)3-s + i·9-s + (−1 − i)13-s − i·23-s − i·25-s + (−1 − i)29-s + 2i·39-s + 2i·41-s − 2·47-s − 49-s + (1 − i)59-s + (−1 + i)69-s − 2i·73-s + (−1 + i)75-s + 81-s + ⋯
L(s)  = 1  + (−1 − i)3-s + i·9-s + (−1 − i)13-s − i·23-s − i·25-s + (−1 − i)29-s + 2i·39-s + 2i·41-s − 2·47-s − 49-s + (1 − i)59-s + (−1 + i)69-s − 2i·73-s + (−1 + i)75-s + 81-s + ⋯

Functional equation

Λ(s)=(1472s/2 ΓC(s) L(s)=((−0.923+0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1472s/2 ΓC(s) L(s)=((−0.923+0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14721472    =    26⋅232^{6} \cdot 23
Sign: −0.923+0.382i-0.923 + 0.382i
Analytic conductor: 0.7346230.734623
Root analytic conductor: 0.8571010.857101
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1472(1425,⋅)\chi_{1472} (1425, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1472, ( :0), −0.923+0.382i)(2,\ 1472,\ (\ :0),\ -0.923 + 0.382i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.47815490060.4781549006
L(12)L(\frac12) ≈\approx 0.47815490060.4781549006
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 1
23 1+iT 1 + iT
good3 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
5 1+iT2 1 + iT^{2}
7 1+T2 1 + T^{2}
11 1+iT2 1 + iT^{2}
13 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
17 1−T2 1 - T^{2}
19 1−iT2 1 - iT^{2}
29 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
31 1+T2 1 + T^{2}
37 1+iT2 1 + iT^{2}
41 1−2iT−T2 1 - 2iT - T^{2}
43 1+iT2 1 + iT^{2}
47 1+2T+T2 1 + 2T + T^{2}
53 1+iT2 1 + iT^{2}
59 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
61 1−iT2 1 - iT^{2}
67 1−iT2 1 - iT^{2}
71 1−T2 1 - T^{2}
73 1+2iT−T2 1 + 2iT - T^{2}
79 1−T2 1 - T^{2}
83 1−iT2 1 - iT^{2}
89 1+T2 1 + 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

−9.540864728123536893037153029622, −8.146934818605474248307196468854, −7.79131131330721310153342056583, −6.71659709475276646863850361219, −6.24395529802743671147969119772, −5.31232219173448931934722221886, −4.54731029416957891172573435628, −3.05080532227281323577501297150, −1.90870223265964331687850674993, −0.42333670295963037680140036137, 1.84990701873601891355084067052, 3.41319010040791130690878703668, 4.27555675899622033603690998351, 5.17504049756295581018714727881, 5.61806620553868395572879005222, 6.82033105735933499790066607771, 7.41861745556457510078669580158, 8.691518392501832098574446649930, 9.572983584122506562908521870706, 9.891766602840267351340748700194

Graph of the ZZ-function along the critical line