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Properties

Label 2-3072-96.77-c0-0-4
Degree 22
Conductor 30723072
Sign 0.980+0.195i0.980 + 0.195i
Analytic cond. 1.533121.53312
Root an. cond. 1.238191.23819
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.382 + 0.923i)3-s + (1.30 − 1.30i)7-s + (−0.707 − 0.707i)9-s + (0.707 + 0.292i)13-s + (−1.30 − 0.541i)19-s + (0.707 + 1.70i)21-s + (0.707 − 0.707i)25-s + (0.923 − 0.382i)27-s − 0.765·31-s + (1.70 − 0.707i)37-s + (−0.541 + 0.541i)39-s + (−0.541 − 1.30i)43-s − 2.41i·49-s + (1 − 0.999i)57-s + (−0.292 + 0.707i)61-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)3-s + (1.30 − 1.30i)7-s + (−0.707 − 0.707i)9-s + (0.707 + 0.292i)13-s + (−1.30 − 0.541i)19-s + (0.707 + 1.70i)21-s + (0.707 − 0.707i)25-s + (0.923 − 0.382i)27-s − 0.765·31-s + (1.70 − 0.707i)37-s + (−0.541 + 0.541i)39-s + (−0.541 − 1.30i)43-s − 2.41i·49-s + (1 − 0.999i)57-s + (−0.292 + 0.707i)61-s + ⋯

Functional equation

Λ(s)=(3072s/2 ΓC(s) L(s)=((0.980+0.195i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3072s/2 ΓC(s) L(s)=((0.980+0.195i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 30723072    =    210⋅32^{10} \cdot 3
Sign: 0.980+0.195i0.980 + 0.195i
Analytic conductor: 1.533121.53312
Root analytic conductor: 1.238191.23819
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3072(2177,⋅)\chi_{3072} (2177, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3072, ( :0), 0.980+0.195i)(2,\ 3072,\ (\ :0),\ 0.980 + 0.195i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2252984531.225298453
L(12)L(\frac12) ≈\approx 1.2252984531.225298453
L(1)L(1) 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+(0.382−0.923i)T 1 + (0.382 - 0.923i)T
good5 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
7 1+(−1.30+1.30i)T−iT2 1 + (-1.30 + 1.30i)T - iT^{2}
11 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
13 1+(−0.707−0.292i)T+(0.707+0.707i)T2 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2}
17 1+T2 1 + T^{2}
19 1+(1.30+0.541i)T+(0.707+0.707i)T2 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2}
23 1−iT2 1 - iT^{2}
29 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
31 1+0.765T+T2 1 + 0.765T + T^{2}
37 1+(−1.70+0.707i)T+(0.707−0.707i)T2 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2}
41 1−iT2 1 - iT^{2}
43 1+(0.541+1.30i)T+(−0.707+0.707i)T2 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2}
47 1+T2 1 + T^{2}
53 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
59 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
61 1+(0.292−0.707i)T+(−0.707−0.707i)T2 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2}
67 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
71 1+iT2 1 + iT^{2}
73 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
79 1−0.765iT−T2 1 - 0.765iT - T^{2}
83 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
89 1+iT2 1 + iT^{2}
97 1+1.41T+T2 1 + 1.41T + 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

−8.718797723963604676414847655392, −8.349459784305606680634166758709, −7.33132605236286392923352648152, −6.60430840550760499705570948533, −5.71121093011620643453870735485, −4.76220741261919322360261060431, −4.28124740498994560565090715496, −3.66028767285828563451165248798, −2.24303292532619186212559181329, −0.887789533032539440528755308945, 1.38130488350925638453659598462, 2.09517378355094455193726919943, 3.05974114651057671698286123068, 4.49179474851744499286368012765, 5.21422670791467994645814252381, 5.96622035073771262403099033054, 6.45452385160577013397034904537, 7.60261226699776712865457861979, 8.182645539777322994567586466522, 8.613351480286552311731129771381

Graph of the ZZ-function along the critical line