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Properties

Label 2-1216-76.31-c1-0-7
Degree 22
Conductor 12161216
Sign −0.417−0.908i-0.417 - 0.908i
Analytic cond. 9.709809.70980
Root an. cond. 3.116053.11605
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  + (0.637 + 1.10i)3-s + (1.60 + 2.77i)5-s + 1.25i·7-s + (0.688 − 1.19i)9-s + 2.11i·11-s + (−2.12 − 1.22i)13-s + (−2.04 + 3.53i)15-s + (0.765 + 1.32i)17-s + (−3.76 + 2.19i)19-s + (−1.37 + 0.796i)21-s + (7.61 + 4.39i)23-s + (−2.64 + 4.57i)25-s + 5.57·27-s + (5.20 + 3.00i)29-s − 7.78·31-s + ⋯
L(s)  = 1  + (0.367 + 0.637i)3-s + (0.717 + 1.24i)5-s + 0.472i·7-s + (0.229 − 0.397i)9-s + 0.636i·11-s + (−0.590 − 0.341i)13-s + (−0.527 + 0.913i)15-s + (0.185 + 0.321i)17-s + (−0.863 + 0.504i)19-s + (−0.301 + 0.173i)21-s + (1.58 + 0.917i)23-s + (−0.528 + 0.914i)25-s + 1.07·27-s + (0.967 + 0.558i)29-s − 1.39·31-s + ⋯

Functional equation

Λ(s)=(1216s/2 ΓC(s) L(s)=((−0.417−0.908i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1216s/2 ΓC(s+1/2) L(s)=((−0.417−0.908i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12161216    =    26⋅192^{6} \cdot 19
Sign: −0.417−0.908i-0.417 - 0.908i
Analytic conductor: 9.709809.70980
Root analytic conductor: 3.116053.11605
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1216(639,⋅)\chi_{1216} (639, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1216, ( :1/2), −0.417−0.908i)(2,\ 1216,\ (\ :1/2),\ -0.417 - 0.908i)

Particular Values

L(1)L(1) ≈\approx 2.0247256752.024725675
L(12)L(\frac12) ≈\approx 2.0247256752.024725675
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
19 1+(3.76−2.19i)T 1 + (3.76 - 2.19i)T
good3 1+(−0.637−1.10i)T+(−1.5+2.59i)T2 1 + (-0.637 - 1.10i)T + (-1.5 + 2.59i)T^{2}
5 1+(−1.60−2.77i)T+(−2.5+4.33i)T2 1 + (-1.60 - 2.77i)T + (-2.5 + 4.33i)T^{2}
7 1−1.25iT−7T2 1 - 1.25iT - 7T^{2}
11 1−2.11iT−11T2 1 - 2.11iT - 11T^{2}
13 1+(2.12+1.22i)T+(6.5+11.2i)T2 1 + (2.12 + 1.22i)T + (6.5 + 11.2i)T^{2}
17 1+(−0.765−1.32i)T+(−8.5+14.7i)T2 1 + (-0.765 - 1.32i)T + (-8.5 + 14.7i)T^{2}
23 1+(−7.61−4.39i)T+(11.5+19.9i)T2 1 + (-7.61 - 4.39i)T + (11.5 + 19.9i)T^{2}
29 1+(−5.20−3.00i)T+(14.5+25.1i)T2 1 + (-5.20 - 3.00i)T + (14.5 + 25.1i)T^{2}
31 1+7.78T+31T2 1 + 7.78T + 31T^{2}
37 1+9.97iT−37T2 1 + 9.97iT - 37T^{2}
41 1+(−1.09+0.631i)T+(20.5−35.5i)T2 1 + (-1.09 + 0.631i)T + (20.5 - 35.5i)T^{2}
43 1+(5.04−2.91i)T+(21.5−37.2i)T2 1 + (5.04 - 2.91i)T + (21.5 - 37.2i)T^{2}
47 1+(6.12+3.53i)T+(23.5+40.7i)T2 1 + (6.12 + 3.53i)T + (23.5 + 40.7i)T^{2}
53 1+(6.18+3.57i)T+(26.5+45.8i)T2 1 + (6.18 + 3.57i)T + (26.5 + 45.8i)T^{2}
59 1+(2.83+4.91i)T+(−29.5+51.0i)T2 1 + (2.83 + 4.91i)T + (-29.5 + 51.0i)T^{2}
61 1+(2.80−4.86i)T+(−30.5−52.8i)T2 1 + (2.80 - 4.86i)T + (-30.5 - 52.8i)T^{2}
67 1+(0.0235−0.0408i)T+(−33.5−58.0i)T2 1 + (0.0235 - 0.0408i)T + (-33.5 - 58.0i)T^{2}
71 1+(−3.12−5.41i)T+(−35.5+61.4i)T2 1 + (-3.12 - 5.41i)T + (-35.5 + 61.4i)T^{2}
73 1+(−0.658−1.13i)T+(−36.5+63.2i)T2 1 + (-0.658 - 1.13i)T + (-36.5 + 63.2i)T^{2}
79 1+(−3.77−6.53i)T+(−39.5+68.4i)T2 1 + (-3.77 - 6.53i)T + (-39.5 + 68.4i)T^{2}
83 1−7.84iT−83T2 1 - 7.84iT - 83T^{2}
89 1+(6.02+3.47i)T+(44.5+77.0i)T2 1 + (6.02 + 3.47i)T + (44.5 + 77.0i)T^{2}
97 1+(−8.51+4.91i)T+(48.5−84.0i)T2 1 + (-8.51 + 4.91i)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

−9.930297564976434082742544978558, −9.396250486259978805584418989875, −8.572156971654281768047756971871, −7.32004513085687527246207507840, −6.77691428508652248315426730802, −5.80904243089063513602267832452, −4.89622299182986509117394761913, −3.66993057422961581593509505151, −2.91037463125627058054135140436, −1.87798175826145707021823497522, 0.839731580083763358103983801992, 1.87957751273009574078253571767, 2.98714430545485521610856607887, 4.65015965061072872611120592445, 4.93922424151873473921896351062, 6.24980003929915021151682406471, 7.00732368273116638523484716669, 7.941962984474611954283387533436, 8.703962697591025673673070488733, 9.237958454183641360794270438557

Graph of the ZZ-function along the critical line