↑

Properties

Label 2-1456-13.9-c1-0-10
Degree 22
Conductor 14561456
Sign −0.0128−0.999i-0.0128 - 0.999i
Analytic cond. 11.626211.6262
Root an. cond. 3.409723.40972
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  − 5-s + (−0.5 − 0.866i)7-s + (1.5 + 2.59i)9-s + (−1 + 1.73i)11-s + (3.5 + 0.866i)13-s + (1.5 + 2.59i)17-s + (−3 − 5.19i)19-s + (−2 + 3.46i)23-s − 4·25-s + (3.5 − 6.06i)29-s − 4·31-s + (0.5 + 0.866i)35-s + (−4.5 + 7.79i)37-s + (−4.5 + 7.79i)41-s + (5 + 8.66i)43-s + ⋯
L(s)  = 1  − 0.447·5-s + (−0.188 − 0.327i)7-s + (0.5 + 0.866i)9-s + (−0.301 + 0.522i)11-s + (0.970 + 0.240i)13-s + (0.363 + 0.630i)17-s + (−0.688 − 1.19i)19-s + (−0.417 + 0.722i)23-s − 0.800·25-s + (0.649 − 1.12i)29-s − 0.718·31-s + (0.0845 + 0.146i)35-s + (−0.739 + 1.28i)37-s + (−0.702 + 1.21i)41-s + (0.762 + 1.32i)43-s + ⋯

Functional equation

Λ(s)=(1456s/2 ΓC(s) L(s)=((−0.0128−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1456s/2 ΓC(s+1/2) L(s)=((−0.0128−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14561456    =    24⋅7⋅132^{4} \cdot 7 \cdot 13
Sign: −0.0128−0.999i-0.0128 - 0.999i
Analytic conductor: 11.626211.6262
Root analytic conductor: 3.409723.40972
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1456(113,⋅)\chi_{1456} (113, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1456, ( :1/2), −0.0128−0.999i)(2,\ 1456,\ (\ :1/2),\ -0.0128 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 1.1903265681.190326568
L(12)L(\frac12) ≈\approx 1.1903265681.190326568
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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
13 1+(−3.5−0.866i)T 1 + (-3.5 - 0.866i)T
good3 1+(−1.5−2.59i)T2 1 + (-1.5 - 2.59i)T^{2}
5 1+T+5T2 1 + T + 5T^{2}
11 1+(1−1.73i)T+(−5.5−9.52i)T2 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2}
17 1+(−1.5−2.59i)T+(−8.5+14.7i)T2 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2}
19 1+(3+5.19i)T+(−9.5+16.4i)T2 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2}
23 1+(2−3.46i)T+(−11.5−19.9i)T2 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2}
29 1+(−3.5+6.06i)T+(−14.5−25.1i)T2 1 + (-3.5 + 6.06i)T + (-14.5 - 25.1i)T^{2}
31 1+4T+31T2 1 + 4T + 31T^{2}
37 1+(4.5−7.79i)T+(−18.5−32.0i)T2 1 + (4.5 - 7.79i)T + (-18.5 - 32.0i)T^{2}
41 1+(4.5−7.79i)T+(−20.5−35.5i)T2 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2}
43 1+(−5−8.66i)T+(−21.5+37.2i)T2 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2}
47 1−2T+47T2 1 - 2T + 47T^{2}
53 1−9T+53T2 1 - 9T + 53T^{2}
59 1+(−7−12.1i)T+(−29.5+51.0i)T2 1 + (-7 - 12.1i)T + (-29.5 + 51.0i)T^{2}
61 1+(−2.5−4.33i)T+(−30.5+52.8i)T2 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2}
67 1+(4−6.92i)T+(−33.5−58.0i)T2 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2}
71 1+(−5−8.66i)T+(−35.5+61.4i)T2 1 + (-5 - 8.66i)T + (-35.5 + 61.4i)T^{2}
73 1+7T+73T2 1 + 7T + 73T^{2}
79 1+2T+79T2 1 + 2T + 79T^{2}
83 1−6T+83T2 1 - 6T + 83T^{2}
89 1+(3−5.19i)T+(−44.5−77.0i)T2 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2}
97 1+(1+1.73i)T+(−48.5+84.0i)T2 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{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

−9.957092576590929367851818577979, −8.784912819514252520236630864563, −8.063469087569654328525366741188, −7.38559778192496063108922249683, −6.55317747396699365239383293319, −5.59314324856627679174299047848, −4.47295037722606286586029578956, −3.94016288269780118022406740548, −2.61643743439669012360120528646, −1.40151883700983408992146203129, 0.49870685824528081504784984928, 2.01801590895761174109505791876, 3.55648629853055582686055585539, 3.82107033005566794003713054272, 5.28790517537526930567765011410, 6.02302168340223156310534432405, 6.85669009648069150113453610028, 7.74811084883029840071587525329, 8.629262595615513338256919224167, 9.114057566145506150334802496250

Graph of the ZZ-function along the critical line