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Properties

Label 2-448-448.349-c0-0-0
Degree 22
Conductor 448448
Sign −0.290−0.956i-0.290 - 0.956i
Analytic cond. 0.2235810.223581
Root an. cond. 0.4728430.472843
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  + (0.382 + 0.923i)2-s + (−0.707 + 0.707i)4-s + (−0.923 + 0.382i)7-s + (−0.923 − 0.382i)8-s + (0.923 + 0.382i)9-s + (0.324 + 1.63i)11-s + (−0.707 − 0.707i)14-s − i·16-s + i·18-s + (−1.38 + 0.923i)22-s + (0.707 − 1.70i)23-s + (−0.382 − 0.923i)25-s + (0.382 − 0.923i)28-s + (0.216 − 1.08i)29-s + (0.923 − 0.382i)32-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)2-s + (−0.707 + 0.707i)4-s + (−0.923 + 0.382i)7-s + (−0.923 − 0.382i)8-s + (0.923 + 0.382i)9-s + (0.324 + 1.63i)11-s + (−0.707 − 0.707i)14-s − i·16-s + i·18-s + (−1.38 + 0.923i)22-s + (0.707 − 1.70i)23-s + (−0.382 − 0.923i)25-s + (0.382 − 0.923i)28-s + (0.216 − 1.08i)29-s + (0.923 − 0.382i)32-s + ⋯

Functional equation

Λ(s)=(448s/2 ΓC(s) L(s)=((−0.290−0.956i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(448s/2 ΓC(s) L(s)=((−0.290−0.956i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 448448    =    26⋅72^{6} \cdot 7
Sign: −0.290−0.956i-0.290 - 0.956i
Analytic conductor: 0.2235810.223581
Root analytic conductor: 0.4728430.472843
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ448(349,⋅)\chi_{448} (349, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 448, ( :0), −0.290−0.956i)(2,\ 448,\ (\ :0),\ -0.290 - 0.956i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.91536279970.9153627997
L(12)L(\frac12) ≈\approx 0.91536279970.9153627997
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+(−0.382−0.923i)T 1 + (-0.382 - 0.923i)T
7 1+(0.923−0.382i)T 1 + (0.923 - 0.382i)T
good3 1+(−0.923−0.382i)T2 1 + (-0.923 - 0.382i)T^{2}
5 1+(0.382+0.923i)T2 1 + (0.382 + 0.923i)T^{2}
11 1+(−0.324−1.63i)T+(−0.923+0.382i)T2 1 + (-0.324 - 1.63i)T + (-0.923 + 0.382i)T^{2}
13 1+(0.382−0.923i)T2 1 + (0.382 - 0.923i)T^{2}
17 1−iT2 1 - iT^{2}
19 1+(−0.382+0.923i)T2 1 + (-0.382 + 0.923i)T^{2}
23 1+(−0.707+1.70i)T+(−0.707−0.707i)T2 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2}
29 1+(−0.216+1.08i)T+(−0.923−0.382i)T2 1 + (-0.216 + 1.08i)T + (-0.923 - 0.382i)T^{2}
31 1+T2 1 + T^{2}
37 1+(0.324+0.216i)T+(0.382+0.923i)T2 1 + (0.324 + 0.216i)T + (0.382 + 0.923i)T^{2}
41 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
43 1+(0.382−0.0761i)T+(0.923−0.382i)T2 1 + (0.382 - 0.0761i)T + (0.923 - 0.382i)T^{2}
47 1−iT2 1 - iT^{2}
53 1+(−0.382−1.92i)T+(−0.923+0.382i)T2 1 + (-0.382 - 1.92i)T + (-0.923 + 0.382i)T^{2}
59 1+(0.382+0.923i)T2 1 + (0.382 + 0.923i)T^{2}
61 1+(−0.923−0.382i)T2 1 + (-0.923 - 0.382i)T^{2}
67 1+(1.92+0.382i)T+(0.923+0.382i)T2 1 + (1.92 + 0.382i)T + (0.923 + 0.382i)T^{2}
71 1+(−1.30+0.541i)T+(0.707−0.707i)T2 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2}
73 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
79 1+(0.541−0.541i)T−iT2 1 + (0.541 - 0.541i)T - iT^{2}
83 1+(−0.382+0.923i)T2 1 + (-0.382 + 0.923i)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

−12.11639916572497631634618615686, −10.35975306681417962045267137553, −9.703128394198623611340275720439, −8.827714643004513568855999051045, −7.66380835195516936024913954204, −6.87535110408835168306397938790, −6.18073434917292392613427763235, −4.77629470737715626755322504908, −4.13307241291556885672749716365, −2.51143894717075304928801970058, 1.27416153341040169943284816662, 3.29279849857578592394624745439, 3.69589403950151495664122088322, 5.20708169082076706543131504249, 6.20107513393605711136734091612, 7.21785965465829034292177828359, 8.726505306528812413204516739893, 9.451934533169859489030986156027, 10.22517330402765310952347166300, 11.13514997682342806357188859709

Graph of the ZZ-function along the critical line