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Properties

Label 2-2448-204.143-c0-0-1
Degree 22
Conductor 24482448
Sign 0.999+0.00538i0.999 + 0.00538i
Analytic cond. 1.221711.22171
Root an. cond. 1.105311.10531
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  + (1.92 − 0.382i)5-s + (−0.541 + 0.541i)13-s + i·17-s + (2.63 − 1.08i)25-s + (0.324 + 1.63i)29-s + (1.08 − 1.63i)37-s + (−1.08 − 0.216i)41-s + (−0.923 − 0.382i)49-s + (−0.707 − 0.292i)53-s + (−1.63 − 0.324i)61-s + (−0.834 + 1.24i)65-s + (−0.382 − 1.92i)73-s + (0.382 + 1.92i)85-s + (0.541 − 0.541i)89-s + (1.08 − 0.216i)97-s + ⋯
L(s)  = 1  + (1.92 − 0.382i)5-s + (−0.541 + 0.541i)13-s + i·17-s + (2.63 − 1.08i)25-s + (0.324 + 1.63i)29-s + (1.08 − 1.63i)37-s + (−1.08 − 0.216i)41-s + (−0.923 − 0.382i)49-s + (−0.707 − 0.292i)53-s + (−1.63 − 0.324i)61-s + (−0.834 + 1.24i)65-s + (−0.382 − 1.92i)73-s + (0.382 + 1.92i)85-s + (0.541 − 0.541i)89-s + (1.08 − 0.216i)97-s + ⋯

Functional equation

Λ(s)=(2448s/2 ΓC(s) L(s)=((0.999+0.00538i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00538i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2448s/2 ΓC(s) L(s)=((0.999+0.00538i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00538i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24482448    =    24⋅32⋅172^{4} \cdot 3^{2} \cdot 17
Sign: 0.999+0.00538i0.999 + 0.00538i
Analytic conductor: 1.221711.22171
Root analytic conductor: 1.105311.10531
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2448(143,⋅)\chi_{2448} (143, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2448, ( :0), 0.999+0.00538i)(2,\ 2448,\ (\ :0),\ 0.999 + 0.00538i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.6682613921.668261392
L(12)L(\frac12) ≈\approx 1.6682613921.668261392
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
3 1 1
17 1−iT 1 - iT
good5 1+(−1.92+0.382i)T+(0.923−0.382i)T2 1 + (-1.92 + 0.382i)T + (0.923 - 0.382i)T^{2}
7 1+(0.923+0.382i)T2 1 + (0.923 + 0.382i)T^{2}
11 1+(0.382−0.923i)T2 1 + (0.382 - 0.923i)T^{2}
13 1+(0.541−0.541i)T−iT2 1 + (0.541 - 0.541i)T - iT^{2}
19 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
23 1+(−0.382+0.923i)T2 1 + (-0.382 + 0.923i)T^{2}
29 1+(−0.324−1.63i)T+(−0.923+0.382i)T2 1 + (-0.324 - 1.63i)T + (-0.923 + 0.382i)T^{2}
31 1+(−0.382−0.923i)T2 1 + (-0.382 - 0.923i)T^{2}
37 1+(−1.08+1.63i)T+(−0.382−0.923i)T2 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2}
41 1+(1.08+0.216i)T+(0.923+0.382i)T2 1 + (1.08 + 0.216i)T + (0.923 + 0.382i)T^{2}
43 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
47 1−iT2 1 - iT^{2}
53 1+(0.707+0.292i)T+(0.707+0.707i)T2 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2}
59 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
61 1+(1.63+0.324i)T+(0.923+0.382i)T2 1 + (1.63 + 0.324i)T + (0.923 + 0.382i)T^{2}
67 1+T2 1 + T^{2}
71 1+(−0.382−0.923i)T2 1 + (-0.382 - 0.923i)T^{2}
73 1+(0.382+1.92i)T+(−0.923+0.382i)T2 1 + (0.382 + 1.92i)T + (-0.923 + 0.382i)T^{2}
79 1+(−0.382+0.923i)T2 1 + (-0.382 + 0.923i)T^{2}
83 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
89 1+(−0.541+0.541i)T−iT2 1 + (-0.541 + 0.541i)T - iT^{2}
97 1+(−1.08+0.216i)T+(0.923−0.382i)T2 1 + (-1.08 + 0.216i)T + (0.923 - 0.382i)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

−9.129732029081890544004737107451, −8.678841197161910598014350925556, −7.54861264013351767836896921779, −6.55980437702882575860074838147, −6.08602310104126069646110004502, −5.22856069132374263722647492126, −4.60847673234576695104990858784, −3.26312736417885952956850399770, −2.13940061779729048441012798485, −1.50493360291353787008980613226, 1.36325453808845905103828391990, 2.51322207374640661923634823954, 2.99282666787585403103213062234, 4.61065365357188190013537413586, 5.26951220694871638713747592808, 6.10968542375919954731786616517, 6.59167729483433782836308829048, 7.53900165000748765268401109112, 8.427352963738548716740190842972, 9.496927067737046169802575015598

Graph of the ZZ-function along the critical line