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Properties

Label 2-3276-91.51-c1-0-23
Degree 22
Conductor 32763276
Sign 0.596+0.802i0.596 + 0.802i
Analytic cond. 26.158926.1589
Root an. cond. 5.114585.11458
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  + (−2.5 + 0.866i)7-s + (−2.5 − 2.59i)13-s + (7.5 + 4.33i)19-s + (−2.5 − 4.33i)25-s + (−1.5 + 0.866i)31-s + (4.5 + 2.59i)37-s − 13·43-s + (5.5 − 4.33i)49-s + (7 − 12.1i)61-s + (13.5 − 7.79i)67-s + (−1.5 + 0.866i)73-s + (8.5 − 14.7i)79-s + (8.5 + 4.33i)91-s − 13.8i·97-s + (6.5 − 11.2i)103-s + ⋯
L(s)  = 1  + (−0.944 + 0.327i)7-s + (−0.693 − 0.720i)13-s + (1.72 + 0.993i)19-s + (−0.5 − 0.866i)25-s + (−0.269 + 0.155i)31-s + (0.739 + 0.427i)37-s − 1.98·43-s + (0.785 − 0.618i)49-s + (0.896 − 1.55i)61-s + (1.64 − 0.952i)67-s + (−0.175 + 0.101i)73-s + (0.956 − 1.65i)79-s + (0.891 + 0.453i)91-s − 1.40i·97-s + (0.640 − 1.10i)103-s + ⋯

Functional equation

Λ(s)=(3276s/2 ΓC(s) L(s)=((0.596+0.802i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3276s/2 ΓC(s+1/2) L(s)=((0.596+0.802i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32763276    =    22⋅32⋅7⋅132^{2} \cdot 3^{2} \cdot 7 \cdot 13
Sign: 0.596+0.802i0.596 + 0.802i
Analytic conductor: 26.158926.1589
Root analytic conductor: 5.114585.11458
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3276(2053,⋅)\chi_{3276} (2053, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3276, ( :1/2), 0.596+0.802i)(2,\ 3276,\ (\ :1/2),\ 0.596 + 0.802i)

Particular Values

L(1)L(1) ≈\approx 1.2712964421.271296442
L(12)L(\frac12) ≈\approx 1.2712964421.271296442
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
7 1+(2.5−0.866i)T 1 + (2.5 - 0.866i)T
13 1+(2.5+2.59i)T 1 + (2.5 + 2.59i)T
good5 1+(2.5+4.33i)T2 1 + (2.5 + 4.33i)T^{2}
11 1+(5.5−9.52i)T2 1 + (5.5 - 9.52i)T^{2}
17 1+(−8.5+14.7i)T2 1 + (-8.5 + 14.7i)T^{2}
19 1+(−7.5−4.33i)T+(9.5+16.4i)T2 1 + (-7.5 - 4.33i)T + (9.5 + 16.4i)T^{2}
23 1+(−11.5−19.9i)T2 1 + (-11.5 - 19.9i)T^{2}
29 1+29T2 1 + 29T^{2}
31 1+(1.5−0.866i)T+(15.5−26.8i)T2 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2}
37 1+(−4.5−2.59i)T+(18.5+32.0i)T2 1 + (-4.5 - 2.59i)T + (18.5 + 32.0i)T^{2}
41 1−41T2 1 - 41T^{2}
43 1+13T+43T2 1 + 13T + 43T^{2}
47 1+(23.5+40.7i)T2 1 + (23.5 + 40.7i)T^{2}
53 1+(−26.5+45.8i)T2 1 + (-26.5 + 45.8i)T^{2}
59 1+(29.5−51.0i)T2 1 + (29.5 - 51.0i)T^{2}
61 1+(−7+12.1i)T+(−30.5−52.8i)T2 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2}
67 1+(−13.5+7.79i)T+(33.5−58.0i)T2 1 + (-13.5 + 7.79i)T + (33.5 - 58.0i)T^{2}
71 1−71T2 1 - 71T^{2}
73 1+(1.5−0.866i)T+(36.5−63.2i)T2 1 + (1.5 - 0.866i)T + (36.5 - 63.2i)T^{2}
79 1+(−8.5+14.7i)T+(−39.5−68.4i)T2 1 + (-8.5 + 14.7i)T + (-39.5 - 68.4i)T^{2}
83 1−83T2 1 - 83T^{2}
89 1+(44.5+77.0i)T2 1 + (44.5 + 77.0i)T^{2}
97 1+13.8iT−97T2 1 + 13.8iT - 97T^{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

−8.408400968853407284542667633519, −7.84753506796081356110517177531, −7.03742142585836537349176230122, −6.24231906404629324340955335536, −5.53141501604077199854894710084, −4.81856231557847379849454396950, −3.55572941978070863045475152341, −3.07750612259779421629013139647, −1.94929097529683185927695874197, −0.48264857543684556610485166282, 0.923929624190572978977919317980, 2.30288426504281922957481550234, 3.22147637567794233208054067250, 3.97667235386509789704582611136, 5.00684708472396509681380948929, 5.65695020548949990029194413178, 6.77173014569794185282399290274, 7.09221912944065934057931731606, 7.88380380386551527209327020779, 8.931145556963242582831228659034

Graph of the ZZ-function along the critical line