↑

Properties

Label 2-448-112.19-c1-0-10
Degree 22
Conductor 448448
Sign 0.486+0.873i0.486 + 0.873i
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  + (0.388 − 1.44i)3-s + (2.81 − 0.755i)5-s + (1.47 − 2.19i)7-s + (0.649 + 0.375i)9-s + (−0.0165 + 0.0619i)11-s + (−3.62 + 3.62i)13-s − 4.37i·15-s + (3.26 − 1.88i)17-s + (−2.13 + 0.572i)19-s + (−2.60 − 2.99i)21-s + (−3.80 + 6.59i)23-s + (3.04 − 1.75i)25-s + (3.97 − 3.97i)27-s + (−4.74 − 4.74i)29-s + (0.329 + 0.570i)31-s + ⋯
L(s)  = 1  + (0.224 − 0.836i)3-s + (1.26 − 0.337i)5-s + (0.558 − 0.829i)7-s + (0.216 + 0.125i)9-s + (−0.00500 + 0.0186i)11-s + (−1.00 + 1.00i)13-s − 1.12i·15-s + (0.790 − 0.456i)17-s + (−0.490 + 0.131i)19-s + (−0.568 − 0.653i)21-s + (−0.793 + 1.37i)23-s + (0.608 − 0.351i)25-s + (0.765 − 0.765i)27-s + (−0.880 − 0.880i)29-s + (0.0591 + 0.102i)31-s + ⋯

Functional equation

Λ(s)=(448s/2 ΓC(s) L(s)=((0.486+0.873i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(448s/2 ΓC(s+1/2) L(s)=((0.486+0.873i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 448448    =    26⋅72^{6} \cdot 7
Sign: 0.486+0.873i0.486 + 0.873i
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), 0.486+0.873i)(2,\ 448,\ (\ :1/2),\ 0.486 + 0.873i)

Particular Values

L(1)L(1) ≈\approx 1.64561−0.966665i1.64561 - 0.966665i
L(12)L(\frac12) ≈\approx 1.64561−0.966665i1.64561 - 0.966665i
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+(−1.47+2.19i)T 1 + (-1.47 + 2.19i)T
good3 1+(−0.388+1.44i)T+(−2.59−1.5i)T2 1 + (-0.388 + 1.44i)T + (-2.59 - 1.5i)T^{2}
5 1+(−2.81+0.755i)T+(4.33−2.5i)T2 1 + (-2.81 + 0.755i)T + (4.33 - 2.5i)T^{2}
11 1+(0.0165−0.0619i)T+(−9.52−5.5i)T2 1 + (0.0165 - 0.0619i)T + (-9.52 - 5.5i)T^{2}
13 1+(3.62−3.62i)T−13iT2 1 + (3.62 - 3.62i)T - 13iT^{2}
17 1+(−3.26+1.88i)T+(8.5−14.7i)T2 1 + (-3.26 + 1.88i)T + (8.5 - 14.7i)T^{2}
19 1+(2.13−0.572i)T+(16.4−9.5i)T2 1 + (2.13 - 0.572i)T + (16.4 - 9.5i)T^{2}
23 1+(3.80−6.59i)T+(−11.5−19.9i)T2 1 + (3.80 - 6.59i)T + (-11.5 - 19.9i)T^{2}
29 1+(4.74+4.74i)T+29iT2 1 + (4.74 + 4.74i)T + 29iT^{2}
31 1+(−0.329−0.570i)T+(−15.5+26.8i)T2 1 + (-0.329 - 0.570i)T + (-15.5 + 26.8i)T^{2}
37 1+(1.08+4.05i)T+(−32.0+18.5i)T2 1 + (1.08 + 4.05i)T + (-32.0 + 18.5i)T^{2}
41 1−7.67T+41T2 1 - 7.67T + 41T^{2}
43 1+(−2.54−2.54i)T+43iT2 1 + (-2.54 - 2.54i)T + 43iT^{2}
47 1+(2.62−4.55i)T+(−23.5−40.7i)T2 1 + (2.62 - 4.55i)T + (-23.5 - 40.7i)T^{2}
53 1+(10.1+2.72i)T+(45.8+26.5i)T2 1 + (10.1 + 2.72i)T + (45.8 + 26.5i)T^{2}
59 1+(2.43+0.652i)T+(51.0+29.5i)T2 1 + (2.43 + 0.652i)T + (51.0 + 29.5i)T^{2}
61 1+(−1.84−6.89i)T+(−52.8+30.5i)T2 1 + (-1.84 - 6.89i)T + (-52.8 + 30.5i)T^{2}
67 1+(5.98+1.60i)T+(58.0+33.5i)T2 1 + (5.98 + 1.60i)T + (58.0 + 33.5i)T^{2}
71 1+1.08T+71T2 1 + 1.08T + 71T^{2}
73 1+(0.0232+0.0402i)T+(−36.5+63.2i)T2 1 + (0.0232 + 0.0402i)T + (-36.5 + 63.2i)T^{2}
79 1+(−15.2−8.80i)T+(39.5+68.4i)T2 1 + (-15.2 - 8.80i)T + (39.5 + 68.4i)T^{2}
83 1+(−5.07−5.07i)T+83iT2 1 + (-5.07 - 5.07i)T + 83iT^{2}
89 1+(4.47−7.75i)T+(−44.5−77.0i)T2 1 + (4.47 - 7.75i)T + (-44.5 - 77.0i)T^{2}
97 1−1.85iT−97T2 1 - 1.85iT - 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

−10.93014058891306765955281170460, −9.771342767736886694462249588162, −9.439323765524127767706072751714, −7.892513713141166187534609324218, −7.41083414936236200860084003534, −6.34297823591747456479816631458, −5.27101841929411522604290219142, −4.17344803731536265950517785781, −2.24396323938358720260189603562, −1.44116291603233712676664213710, 1.99250972292863291790586725309, 3.09103936536382977275518791764, 4.60951470310681683213160848518, 5.49519210783107814727457563999, 6.33797129393659660750219714636, 7.71753960744078133504541915878, 8.763739798634114832109178205054, 9.624357804587167066909121597964, 10.22781402827050933289413307723, 10.87292901868003626916821075153

Graph of the ZZ-function along the critical line