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Properties

Label 2-1332-444.119-c0-0-1
Degree 22
Conductor 13321332
Sign −0.232+0.972i-0.232 + 0.972i
Analytic cond. 0.6647540.664754
Root an. cond. 0.8153240.815324
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  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.448 − 1.67i)5-s + (0.707 − 0.707i)8-s + 1.73·10-s + (−1.36 + 0.366i)13-s + (0.500 + 0.866i)16-s + (−1.67 − 0.448i)17-s + (−0.448 + 1.67i)20-s + (−1.73 + 1.00i)25-s − 1.41i·26-s + (−1.22 − 1.22i)29-s + (−0.965 + 0.258i)32-s + (0.866 − 1.50i)34-s + (0.866 + 0.5i)37-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.448 − 1.67i)5-s + (0.707 − 0.707i)8-s + 1.73·10-s + (−1.36 + 0.366i)13-s + (0.500 + 0.866i)16-s + (−1.67 − 0.448i)17-s + (−0.448 + 1.67i)20-s + (−1.73 + 1.00i)25-s − 1.41i·26-s + (−1.22 − 1.22i)29-s + (−0.965 + 0.258i)32-s + (0.866 − 1.50i)34-s + (0.866 + 0.5i)37-s + ⋯

Functional equation

Λ(s)=(1332s/2 ΓC(s) L(s)=((−0.232+0.972i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1332s/2 ΓC(s) L(s)=((−0.232+0.972i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13321332    =    22⋅32⋅372^{2} \cdot 3^{2} \cdot 37
Sign: −0.232+0.972i-0.232 + 0.972i
Analytic conductor: 0.6647540.664754
Root analytic conductor: 0.8153240.815324
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1332(1007,⋅)\chi_{1332} (1007, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1332, ( :0), −0.232+0.972i)(2,\ 1332,\ (\ :0),\ -0.232 + 0.972i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.37693647250.3769364725
L(12)L(\frac12) ≈\approx 0.37693647250.3769364725
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+(0.258−0.965i)T 1 + (0.258 - 0.965i)T
3 1 1
37 1+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
good5 1+(0.448+1.67i)T+(−0.866+0.5i)T2 1 + (0.448 + 1.67i)T + (-0.866 + 0.5i)T^{2}
7 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1−T2 1 - T^{2}
13 1+(1.36−0.366i)T+(0.866−0.5i)T2 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2}
17 1+(1.67+0.448i)T+(0.866+0.5i)T2 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2}
19 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
23 1+iT2 1 + iT^{2}
29 1+(1.22+1.22i)T+iT2 1 + (1.22 + 1.22i)T + iT^{2}
31 1+iT2 1 + iT^{2}
41 1+(0.258−0.448i)T+(−0.5−0.866i)T2 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2}
43 1−iT2 1 - iT^{2}
47 1+T2 1 + T^{2}
53 1+(−1.22+0.707i)T+(0.5−0.866i)T2 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2}
59 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
61 1+(0.5+1.86i)T+(−0.866+0.5i)T2 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2}
67 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
71 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
73 1−T2 1 - T^{2}
79 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
83 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
89 1+(0.258−0.965i)T+(−0.866−0.5i)T2 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2}
97 1+(0.366+0.366i)T+iT2 1 + (0.366 + 0.366i)T + iT^{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

−9.407150202018582992376092508503, −8.710462397117497720031738115122, −8.010002189661129383652961922912, −7.28528473319063735079488556948, −6.35254655164834961716952533002, −5.22615856690990462996530383655, −4.70605981670910318006786091409, −4.03557955053383243162936116963, −2.00121290385549564476465688502, −0.32884731423221698640502002362, 2.15025920554857763288385247919, 2.82085664481829600162904896979, 3.77895612041739039741648567862, 4.65987313192819089878180434369, 5.94090146894775108878910821304, 7.19263550311315262022211589683, 7.42176654464588595858812208694, 8.599221211509861176245769951979, 9.432606590090073249489692115928, 10.30244033008582094153941791751

Graph of the ZZ-function along the critical line