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Properties

Label 2-6e4-9.4-c1-0-11
Degree 22
Conductor 12961296
Sign 0.939+0.342i0.939 + 0.342i
Analytic cond. 10.348610.3486
Root an. cond. 3.216923.21692
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)5-s + (2 + 3.46i)11-s + (1 − 1.73i)13-s + 2·17-s + 4·19-s + (−4 + 6.92i)23-s + (0.500 + 0.866i)25-s + (−3 − 5.19i)29-s + (4 − 6.92i)31-s + 6·37-s + (3 − 5.19i)41-s + (2 + 3.46i)43-s + (3.5 − 6.06i)49-s − 2·53-s + 7.99·55-s + ⋯
L(s)  = 1  + (0.447 − 0.774i)5-s + (0.603 + 1.04i)11-s + (0.277 − 0.480i)13-s + 0.485·17-s + 0.917·19-s + (−0.834 + 1.44i)23-s + (0.100 + 0.173i)25-s + (−0.557 − 0.964i)29-s + (0.718 − 1.24i)31-s + 0.986·37-s + (0.468 − 0.811i)41-s + (0.304 + 0.528i)43-s + (0.5 − 0.866i)49-s − 0.274·53-s + 1.07·55-s + ⋯

Functional equation

Λ(s)=(1296s/2 ΓC(s) L(s)=((0.939+0.342i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1296s/2 ΓC(s+1/2) L(s)=((0.939+0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12961296    =    24⋅342^{4} \cdot 3^{4}
Sign: 0.939+0.342i0.939 + 0.342i
Analytic conductor: 10.348610.3486
Root analytic conductor: 3.216923.21692
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1296(433,⋅)\chi_{1296} (433, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1296, ( :1/2), 0.939+0.342i)(2,\ 1296,\ (\ :1/2),\ 0.939 + 0.342i)

Particular Values

L(1)L(1) ≈\approx 1.9465368421.946536842
L(12)L(\frac12) ≈\approx 1.9465368421.946536842
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
3 1 1
good5 1+(−1+1.73i)T+(−2.5−4.33i)T2 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2}
7 1+(−3.5+6.06i)T2 1 + (-3.5 + 6.06i)T^{2}
11 1+(−2−3.46i)T+(−5.5+9.52i)T2 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2}
13 1+(−1+1.73i)T+(−6.5−11.2i)T2 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2}
17 1−2T+17T2 1 - 2T + 17T^{2}
19 1−4T+19T2 1 - 4T + 19T^{2}
23 1+(4−6.92i)T+(−11.5−19.9i)T2 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2}
29 1+(3+5.19i)T+(−14.5+25.1i)T2 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2}
31 1+(−4+6.92i)T+(−15.5−26.8i)T2 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2}
37 1−6T+37T2 1 - 6T + 37T^{2}
41 1+(−3+5.19i)T+(−20.5−35.5i)T2 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2}
43 1+(−2−3.46i)T+(−21.5+37.2i)T2 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2}
47 1+(−23.5+40.7i)T2 1 + (-23.5 + 40.7i)T^{2}
53 1+2T+53T2 1 + 2T + 53T^{2}
59 1+(−2+3.46i)T+(−29.5−51.0i)T2 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2}
61 1+(−1−1.73i)T+(−30.5+52.8i)T2 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2}
67 1+(2−3.46i)T+(−33.5−58.0i)T2 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1−10T+73T2 1 - 10T + 73T^{2}
79 1+(4+6.92i)T+(−39.5+68.4i)T2 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2}
83 1+(2+3.46i)T+(−41.5+71.8i)T2 1 + (2 + 3.46i)T + (-41.5 + 71.8i)T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1+(1+1.73i)T+(−48.5+84.0i)T2 1 + (1 + 1.73i)T + (-48.5 + 84.0i)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

−9.693080612129533857606690196559, −8.983782134721568185013332611342, −7.85386502371059481741805053357, −7.37971193477624688681199347483, −6.06476310441097639881411943699, −5.50723316545421681844032654982, −4.49322110001952021370024627619, −3.59357217927929828770270182953, −2.15828077194064114769502294055, −1.06535465090911142508717297971, 1.15379523108787770338247329142, 2.61186409721887211558019447469, 3.43926254724106768493647356392, 4.52768910472864278499460128736, 5.75833180442213004100776537015, 6.35430308442182449281840083855, 7.09746766904641840388238040625, 8.162894979831527448249214535997, 8.897857964504639591692999712761, 9.739436916257339705893562759834

Graph of the ZZ-function along the critical line