↑

Properties

Label 2-448-112.19-c1-0-12
Degree 22
Conductor 448448
Sign −0.569+0.821i-0.569 + 0.821i
Analytic cond. 3.577293.57729
Root an. cond. 1.891371.89137
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.825 − 3.08i)3-s + (1.86 − 0.501i)5-s + (−2.17 + 1.49i)7-s + (−6.21 − 3.58i)9-s + (0.944 − 3.52i)11-s + (0.372 − 0.372i)13-s − 6.17i·15-s + (2.39 − 1.38i)17-s + (−1.60 + 0.431i)19-s + (2.82 + 7.95i)21-s + (1.27 − 2.20i)23-s + (−1.08 + 0.626i)25-s + (−9.41 + 9.41i)27-s + (2.14 + 2.14i)29-s + (2.74 + 4.75i)31-s + ⋯
L(s)  = 1  + (0.476 − 1.77i)3-s + (0.836 − 0.224i)5-s + (−0.823 + 0.566i)7-s + (−2.07 − 1.19i)9-s + (0.284 − 1.06i)11-s + (0.103 − 0.103i)13-s − 1.59i·15-s + (0.582 − 0.336i)17-s + (−0.369 + 0.0989i)19-s + (0.615 + 1.73i)21-s + (0.265 − 0.459i)23-s + (−0.216 + 0.125i)25-s + (−1.81 + 1.81i)27-s + (0.397 + 0.397i)29-s + (0.493 + 0.854i)31-s + ⋯

Functional equation

Λ(s)=(448s/2 ΓC(s) L(s)=((−0.569+0.821i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(448s/2 ΓC(s+1/2) L(s)=((−0.569+0.821i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 448448    =    26⋅72^{6} \cdot 7
Sign: −0.569+0.821i-0.569 + 0.821i
Analytic conductor: 3.577293.57729
Root analytic conductor: 1.891371.89137
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ448(47,⋅)\chi_{448} (47, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 448, ( :1/2), −0.569+0.821i)(2,\ 448,\ (\ :1/2),\ -0.569 + 0.821i)

Particular Values

L(1)L(1) ≈\approx 0.749608−1.43140i0.749608 - 1.43140i
L(12)L(\frac12) ≈\approx 0.749608−1.43140i0.749608 - 1.43140i
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
7 1+(2.17−1.49i)T 1 + (2.17 - 1.49i)T
good3 1+(−0.825+3.08i)T+(−2.59−1.5i)T2 1 + (-0.825 + 3.08i)T + (-2.59 - 1.5i)T^{2}
5 1+(−1.86+0.501i)T+(4.33−2.5i)T2 1 + (-1.86 + 0.501i)T + (4.33 - 2.5i)T^{2}
11 1+(−0.944+3.52i)T+(−9.52−5.5i)T2 1 + (-0.944 + 3.52i)T + (-9.52 - 5.5i)T^{2}
13 1+(−0.372+0.372i)T−13iT2 1 + (-0.372 + 0.372i)T - 13iT^{2}
17 1+(−2.39+1.38i)T+(8.5−14.7i)T2 1 + (-2.39 + 1.38i)T + (8.5 - 14.7i)T^{2}
19 1+(1.60−0.431i)T+(16.4−9.5i)T2 1 + (1.60 - 0.431i)T + (16.4 - 9.5i)T^{2}
23 1+(−1.27+2.20i)T+(−11.5−19.9i)T2 1 + (-1.27 + 2.20i)T + (-11.5 - 19.9i)T^{2}
29 1+(−2.14−2.14i)T+29iT2 1 + (-2.14 - 2.14i)T + 29iT^{2}
31 1+(−2.74−4.75i)T+(−15.5+26.8i)T2 1 + (-2.74 - 4.75i)T + (-15.5 + 26.8i)T^{2}
37 1+(1.21+4.53i)T+(−32.0+18.5i)T2 1 + (1.21 + 4.53i)T + (-32.0 + 18.5i)T^{2}
41 1−3.95T+41T2 1 - 3.95T + 41T^{2}
43 1+(−4.29−4.29i)T+43iT2 1 + (-4.29 - 4.29i)T + 43iT^{2}
47 1+(1.85−3.21i)T+(−23.5−40.7i)T2 1 + (1.85 - 3.21i)T + (-23.5 - 40.7i)T^{2}
53 1+(−6.49−1.74i)T+(45.8+26.5i)T2 1 + (-6.49 - 1.74i)T + (45.8 + 26.5i)T^{2}
59 1+(8.21+2.20i)T+(51.0+29.5i)T2 1 + (8.21 + 2.20i)T + (51.0 + 29.5i)T^{2}
61 1+(−0.101−0.378i)T+(−52.8+30.5i)T2 1 + (-0.101 - 0.378i)T + (-52.8 + 30.5i)T^{2}
67 1+(−13.8−3.72i)T+(58.0+33.5i)T2 1 + (-13.8 - 3.72i)T + (58.0 + 33.5i)T^{2}
71 1−8.98T+71T2 1 - 8.98T + 71T^{2}
73 1+(−0.766−1.32i)T+(−36.5+63.2i)T2 1 + (-0.766 - 1.32i)T + (-36.5 + 63.2i)T^{2}
79 1+(3.47+2.00i)T+(39.5+68.4i)T2 1 + (3.47 + 2.00i)T + (39.5 + 68.4i)T^{2}
83 1+(−3.67−3.67i)T+83iT2 1 + (-3.67 - 3.67i)T + 83iT^{2}
89 1+(−3.35+5.80i)T+(−44.5−77.0i)T2 1 + (-3.35 + 5.80i)T + (-44.5 - 77.0i)T^{2}
97 1+4.52iT−97T2 1 + 4.52iT - 97T^{2}
show more
show less
   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

−10.94806187644290074630371056740, −9.568535039546818403242078005634, −8.832141828479618833179332278346, −8.109070403016293250485054829846, −6.93995218844565540074400126468, −6.19596576522618340101671595234, −5.57308652822555918215716871773, −3.27925773316531355267714158729, −2.37208234623799393794837371889, −1.00309516965368094421155969602, 2.42853122062574701588846221972, 3.64297194469878309189037133644, 4.42701267047640422797176173840, 5.56868580727584080192673432194, 6.60661807806550390834578723292, 7.962748295670356628139893769386, 9.174118892927537380343956129656, 9.779213624448120447901704480633, 10.15526362033631780053828821098, 10.97236702384040227315610871419

Graph of the ZZ-function along the critical line