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Properties

Label 2-2664-2664.787-c0-0-0
Degree 22
Conductor 26642664
Sign 0.200−0.979i0.200 - 0.979i
Analytic cond. 1.329501.32950
Root an. cond. 1.153041.15304
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  + 2-s − 3-s + 4-s + 2i·5-s − 6-s + (0.866 + 0.5i)7-s + 8-s + 9-s + 2i·10-s + (0.5 + 0.866i)11-s − 12-s + (0.866 + 0.5i)14-s − 2i·15-s + 16-s + (0.5 − 0.866i)17-s + 18-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s + 2i·5-s − 6-s + (0.866 + 0.5i)7-s + 8-s + 9-s + 2i·10-s + (0.5 + 0.866i)11-s − 12-s + (0.866 + 0.5i)14-s − 2i·15-s + 16-s + (0.5 − 0.866i)17-s + 18-s + ⋯

Functional equation

Λ(s)=(2664s/2 ΓC(s) L(s)=((0.200−0.979i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2664s/2 ΓC(s) L(s)=((0.200−0.979i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26642664    =    23⋅32⋅372^{3} \cdot 3^{2} \cdot 37
Sign: 0.200−0.979i0.200 - 0.979i
Analytic conductor: 1.329501.32950
Root analytic conductor: 1.153041.15304
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2664(787,⋅)\chi_{2664} (787, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2664, ( :0), 0.200−0.979i)(2,\ 2664,\ (\ :0),\ 0.200 - 0.979i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.9715972681.971597268
L(12)L(\frac12) ≈\approx 1.9715972681.971597268
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−T 1 - T
3 1+T 1 + T
37 1−iT 1 - iT
good5 1−2iT−T2 1 - 2iT - T^{2}
7 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
11 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
13 1−T2 1 - T^{2}
17 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
19 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
23 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
29 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
31 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
41 1+T2 1 + T^{2}
43 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
47 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
53 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
59 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
61 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
67 1+T2 1 + T^{2}
71 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
73 1+T2 1 + T^{2}
79 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
83 1+T2 1 + T^{2}
89 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
97 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)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

−9.576651865487779115605867005167, −7.984838143090963376315065702421, −7.27004219840947574697427896619, −6.74080170838988257635323036004, −6.19943840523013521737338107274, −5.34574614761706370754907764372, −4.53901994322779646355280884674, −3.73203585354155122177891479131, −2.57006299835687468878428135285, −1.92236458969823212868425457315, 1.22097630734855354938861621211, 1.64559591116499920025554879715, 3.81585275138086549106517932102, 4.21994243043075635112543811657, 4.96506311189518886704310911351, 5.75880407151156732546944751865, 5.99145571230371055501214239233, 7.31278641324131355938514636552, 8.063048165179789545291240643804, 8.647561612074441355826861530655

Graph of the ZZ-function along the critical line