↑

Properties

Label 2-448-112.109-c1-0-4
Degree 22
Conductor 448448
Sign 0.378−0.925i0.378 - 0.925i
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  + (0.0543 − 0.0145i)3-s + (1.25 + 0.337i)5-s + (−0.230 + 2.63i)7-s + (−2.59 + 1.49i)9-s + (0.402 + 1.50i)11-s + (1.59 + 1.59i)13-s + 0.0733·15-s + (1.46 − 2.54i)17-s + (−2.05 + 7.65i)19-s + (0.0258 + 0.146i)21-s + (3.91 − 2.26i)23-s + (−2.86 − 1.65i)25-s + (−0.238 + 0.238i)27-s + (2.06 + 2.06i)29-s + (3.14 − 5.43i)31-s + ⋯
L(s)  = 1  + (0.0313 − 0.00841i)3-s + (0.562 + 0.150i)5-s + (−0.0872 + 0.996i)7-s + (−0.865 + 0.499i)9-s + (0.121 + 0.453i)11-s + (0.442 + 0.442i)13-s + 0.0189·15-s + (0.356 − 0.617i)17-s + (−0.470 + 1.75i)19-s + (0.00564 + 0.0320i)21-s + (0.816 − 0.471i)23-s + (−0.572 − 0.330i)25-s + (−0.0459 + 0.0459i)27-s + (0.382 + 0.382i)29-s + (0.564 − 0.976i)31-s + ⋯

Functional equation

Λ(s)=(448s/2 ΓC(s) L(s)=((0.378−0.925i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(448s/2 ΓC(s+1/2) L(s)=((0.378−0.925i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 448448    =    26⋅72^{6} \cdot 7
Sign: 0.378−0.925i0.378 - 0.925i
Analytic conductor: 3.577293.57729
Root analytic conductor: 1.891371.89137
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ448(81,⋅)\chi_{448} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 448, ( :1/2), 0.378−0.925i)(2,\ 448,\ (\ :1/2),\ 0.378 - 0.925i)

Particular Values

L(1)L(1) ≈\approx 1.16017+0.779098i1.16017 + 0.779098i
L(12)L(\frac12) ≈\approx 1.16017+0.779098i1.16017 + 0.779098i
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+(0.230−2.63i)T 1 + (0.230 - 2.63i)T
good3 1+(−0.0543+0.0145i)T+(2.59−1.5i)T2 1 + (-0.0543 + 0.0145i)T + (2.59 - 1.5i)T^{2}
5 1+(−1.25−0.337i)T+(4.33+2.5i)T2 1 + (-1.25 - 0.337i)T + (4.33 + 2.5i)T^{2}
11 1+(−0.402−1.50i)T+(−9.52+5.5i)T2 1 + (-0.402 - 1.50i)T + (-9.52 + 5.5i)T^{2}
13 1+(−1.59−1.59i)T+13iT2 1 + (-1.59 - 1.59i)T + 13iT^{2}
17 1+(−1.46+2.54i)T+(−8.5−14.7i)T2 1 + (-1.46 + 2.54i)T + (-8.5 - 14.7i)T^{2}
19 1+(2.05−7.65i)T+(−16.4−9.5i)T2 1 + (2.05 - 7.65i)T + (-16.4 - 9.5i)T^{2}
23 1+(−3.91+2.26i)T+(11.5−19.9i)T2 1 + (-3.91 + 2.26i)T + (11.5 - 19.9i)T^{2}
29 1+(−2.06−2.06i)T+29iT2 1 + (-2.06 - 2.06i)T + 29iT^{2}
31 1+(−3.14+5.43i)T+(−15.5−26.8i)T2 1 + (-3.14 + 5.43i)T + (-15.5 - 26.8i)T^{2}
37 1+(−5.24−1.40i)T+(32.0+18.5i)T2 1 + (-5.24 - 1.40i)T + (32.0 + 18.5i)T^{2}
41 1−7.34iT−41T2 1 - 7.34iT - 41T^{2}
43 1+(−1.99+1.99i)T−43iT2 1 + (-1.99 + 1.99i)T - 43iT^{2}
47 1+(0.979+1.69i)T+(−23.5+40.7i)T2 1 + (0.979 + 1.69i)T + (-23.5 + 40.7i)T^{2}
53 1+(−3.00−11.2i)T+(−45.8+26.5i)T2 1 + (-3.00 - 11.2i)T + (-45.8 + 26.5i)T^{2}
59 1+(0.793+2.96i)T+(−51.0+29.5i)T2 1 + (0.793 + 2.96i)T + (-51.0 + 29.5i)T^{2}
61 1+(−2.60+9.72i)T+(−52.8−30.5i)T2 1 + (-2.60 + 9.72i)T + (-52.8 - 30.5i)T^{2}
67 1+(−2.11+0.566i)T+(58.0−33.5i)T2 1 + (-2.11 + 0.566i)T + (58.0 - 33.5i)T^{2}
71 1+7.26iT−71T2 1 + 7.26iT - 71T^{2}
73 1+(12.2+7.06i)T+(36.5+63.2i)T2 1 + (12.2 + 7.06i)T + (36.5 + 63.2i)T^{2}
79 1+(−0.961−1.66i)T+(−39.5+68.4i)T2 1 + (-0.961 - 1.66i)T + (-39.5 + 68.4i)T^{2}
83 1+(8.82+8.82i)T+83iT2 1 + (8.82 + 8.82i)T + 83iT^{2}
89 1+(−11.5+6.66i)T+(44.5−77.0i)T2 1 + (-11.5 + 6.66i)T + (44.5 - 77.0i)T^{2}
97 1+9.69T+97T2 1 + 9.69T + 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

−11.36920791921252272048517372897, −10.27933399437833315029988367827, −9.474041421761467883509920649436, −8.587201586206714419174128634445, −7.76804369886177934922162283384, −6.29199778795832142127258683378, −5.79179799498479529277664662727, −4.59682031197621602757501035244, −3.00840835912049952212169895857, −1.95274094742625830783970342797, 0.913751241003020627602320336292, 2.82607986582602909136330068322, 3.93724665807768352546479478549, 5.28414841606152329692172155747, 6.22695213749752950253732475765, 7.13294545872250519752169975139, 8.362340256723341320359231513310, 9.075371335996337752682405324205, 10.07129549790269213602797091472, 10.95279145871707395063051929754

Graph of the ZZ-function along the critical line