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Properties

Label 2-6e4-9.4-c1-0-18
Degree 22
Conductor 12961296
Sign 0.173+0.984i0.173 + 0.984i
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  + (1.5 − 2.59i)5-s + (−0.5 − 0.866i)7-s + (1.5 + 2.59i)11-s + (2 − 3.46i)13-s − 2·19-s + (3 − 5.19i)23-s + (−2 − 3.46i)25-s + (3 + 5.19i)29-s + (2.5 − 4.33i)31-s − 3·35-s + 2·37-s + (−3 + 5.19i)41-s + (−5 − 8.66i)43-s + (−3 − 5.19i)47-s + (3 − 5.19i)49-s + ⋯
L(s)  = 1  + (0.670 − 1.16i)5-s + (−0.188 − 0.327i)7-s + (0.452 + 0.783i)11-s + (0.554 − 0.960i)13-s − 0.458·19-s + (0.625 − 1.08i)23-s + (−0.400 − 0.692i)25-s + (0.557 + 0.964i)29-s + (0.449 − 0.777i)31-s − 0.507·35-s + 0.328·37-s + (−0.468 + 0.811i)41-s + (−0.762 − 1.32i)43-s + (−0.437 − 0.757i)47-s + (0.428 − 0.742i)49-s + ⋯

Functional equation

Λ(s)=(1296s/2 ΓC(s) L(s)=((0.173+0.984i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1296s/2 ΓC(s+1/2) L(s)=((0.173+0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12961296    =    24⋅342^{4} \cdot 3^{4}
Sign: 0.173+0.984i0.173 + 0.984i
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), 0.173+0.984i)(2,\ 1296,\ (\ :1/2),\ 0.173 + 0.984i)

Particular Values

L(1)L(1) ≈\approx 1.8227448641.822744864
L(12)L(\frac12) ≈\approx 1.8227448641.822744864
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+(−1.5+2.59i)T+(−2.5−4.33i)T2 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2}
7 1+(0.5+0.866i)T+(−3.5+6.06i)T2 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2}
11 1+(−1.5−2.59i)T+(−5.5+9.52i)T2 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2}
13 1+(−2+3.46i)T+(−6.5−11.2i)T2 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2}
17 1+17T2 1 + 17T^{2}
19 1+2T+19T2 1 + 2T + 19T^{2}
23 1+(−3+5.19i)T+(−11.5−19.9i)T2 1 + (-3 + 5.19i)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+(−2.5+4.33i)T+(−15.5−26.8i)T2 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2}
37 1−2T+37T2 1 - 2T + 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+(5+8.66i)T+(−21.5+37.2i)T2 1 + (5 + 8.66i)T + (-21.5 + 37.2i)T^{2}
47 1+(3+5.19i)T+(−23.5+40.7i)T2 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2}
53 1+9T+53T2 1 + 9T + 53T^{2}
59 1+(6−10.3i)T+(−29.5−51.0i)T2 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2}
61 1+(4+6.92i)T+(−30.5+52.8i)T2 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2}
67 1+(−7+12.1i)T+(−33.5−58.0i)T2 1 + (-7 + 12.1i)T + (-33.5 - 58.0i)T^{2}
71 1+71T2 1 + 71T^{2}
73 1+7T+73T2 1 + 7T + 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+(−1.5−2.59i)T+(−41.5+71.8i)T2 1 + (-1.5 - 2.59i)T + (-41.5 + 71.8i)T^{2}
89 1−18T+89T2 1 - 18T + 89T^{2}
97 1+(−0.5−0.866i)T+(−48.5+84.0i)T2 1 + (-0.5 - 0.866i)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

−9.435276958998546136666193570814, −8.702492818396631087295748262906, −8.063439338348438485884420609619, −6.89098322128392487308365037656, −6.16009480359902246842373624528, −5.11651406377277662159806782209, −4.53568080039930419362873439029, −3.32954375323983113209664340505, −1.94029362432280859156290811229, −0.808117356047826066913450692885, 1.54328551351828958111255489604, 2.75555679847085373060118928734, 3.52973558185152318059097053045, 4.74319571739044989544841252986, 6.14468672509763761737082098870, 6.25727561079850022886066921031, 7.20919233048888620624394378167, 8.301573397976511241873904350283, 9.148643611844237141857628531640, 9.777757741237360002339327334269

Graph of the ZZ-function along the critical line