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Properties

Label 2-2128-532.75-c0-0-4
Degree 22
Conductor 21282128
Sign 0.605−0.795i0.605 - 0.795i
Analytic cond. 1.062011.06201
Root an. cond. 1.030531.03053
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  + (1.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)9-s + (1.5 − 0.866i)11-s + (0.5 − 0.866i)19-s + (−1.5 − 0.866i)23-s + (1 + 1.73i)25-s + (−1.5 + 0.866i)35-s + 1.73i·43-s + (−1.5 + 0.866i)45-s + (−0.5 + 0.866i)47-s + (−0.499 − 0.866i)49-s + 3·55-s + (−1.5 − 0.866i)61-s + (−0.499 − 0.866i)63-s + ⋯
L(s)  = 1  + (1.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)9-s + (1.5 − 0.866i)11-s + (0.5 − 0.866i)19-s + (−1.5 − 0.866i)23-s + (1 + 1.73i)25-s + (−1.5 + 0.866i)35-s + 1.73i·43-s + (−1.5 + 0.866i)45-s + (−0.5 + 0.866i)47-s + (−0.499 − 0.866i)49-s + 3·55-s + (−1.5 − 0.866i)61-s + (−0.499 − 0.866i)63-s + ⋯

Functional equation

Λ(s)=(2128s/2 ΓC(s) L(s)=((0.605−0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2128s/2 ΓC(s) L(s)=((0.605−0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21282128    =    24⋅7⋅192^{4} \cdot 7 \cdot 19
Sign: 0.605−0.795i0.605 - 0.795i
Analytic conductor: 1.062011.06201
Root analytic conductor: 1.030531.03053
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2128(607,⋅)\chi_{2128} (607, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2128, ( :0), 0.605−0.795i)(2,\ 2128,\ (\ :0),\ 0.605 - 0.795i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4866793211.486679321
L(12)L(\frac12) ≈\approx 1.4866793211.486679321
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
7 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
19 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good3 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
5 1+(−1.5−0.866i)T+(0.5+0.866i)T2 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2}
11 1+(−1.5+0.866i)T+(0.5−0.866i)T2 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2}
13 1+T2 1 + T^{2}
17 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
23 1+(1.5+0.866i)T+(0.5+0.866i)T2 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2}
29 1−T2 1 - T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
41 1+T2 1 + T^{2}
43 1−1.73iT−T2 1 - 1.73iT - T^{2}
47 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
53 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(1.5+0.866i)T+(0.5+0.866i)T2 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2}
67 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
71 1+T2 1 + T^{2}
73 1+(−1.5+0.866i)T+(0.5−0.866i)T2 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2}
79 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
83 1+T+T2 1 + T + T^{2}
89 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
97 1+T2 1 + 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

−9.421584615111340830512321215627, −8.830654638006002333635792837989, −7.932388157250182007583918749478, −6.68239033022775538223791710951, −6.19356564175547818110603539975, −5.77093257515846501185404354096, −4.71215673595669593516441236667, −3.27891043958527578360939212951, −2.61463658342847155891111444516, −1.72080273339896398541295768904, 1.18230460808785983394956349765, 1.98778001504534548398865043734, 3.55317679433870760296363359254, 4.16926382172128054001661105122, 5.33911680886528468558921014472, 6.08451828908086746233396723287, 6.60070047556043472933429782229, 7.52554980726921347994747031501, 8.708506787600451563904095291912, 9.262675324269625945857583643064

Graph of the ZZ-function along the critical line