Properties

Label 2-2128-532.75-c0-0-5
Degree 22
Conductor 21282128
Sign 0.386+0.922i0.386 + 0.922i
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 + 7-s + (−0.5 + 0.866i)9-s + (1.5 − 0.866i)17-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 + (1 − 1.73i)47-s + 49-s + (−0.5 + 0.866i)63-s + (−0.499 − 0.866i)81-s − 83-s + ⋯
L(s)  = 1  + (−1.5 − 0.866i)5-s + 7-s + (−0.5 + 0.866i)9-s + (1.5 − 0.866i)17-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 + (1 − 1.73i)47-s + 49-s + (−0.5 + 0.866i)63-s + (−0.499 − 0.866i)81-s − 83-s + ⋯

Functional equation

Λ(s)=(2128s/2ΓC(s)L(s)=((0.386+0.922i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2128s/2ΓC(s)L(s)=((0.386+0.922i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21282128    =    247192^{4} \cdot 7 \cdot 19
Sign: 0.386+0.922i0.386 + 0.922i
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), 0.386+0.922i)(2,\ 2128,\ (\ :0),\ 0.386 + 0.922i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.90132831350.9013283135
L(12)L(\frac12) \approx 0.90132831350.9013283135
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
7 1T 1 - T
19 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good3 1+(0.50.866i)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+(0.50.866i)T2 1 + (0.5 - 0.866i)T^{2}
13 1+T2 1 + T^{2}
17 1+(1.5+0.866i)T+(0.50.866i)T2 1 + (-1.5 + 0.866i)T + (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 1T2 1 - T^{2}
31 1+(0.50.866i)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+1.73iTT2 1 + 1.73iT - T^{2}
47 1+(1+1.73i)T+(0.50.866i)T2 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2}
53 1+(0.50.866i)T2 1 + (0.5 - 0.866i)T^{2}
59 1+(0.50.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(0.5+0.866i)T2 1 + (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+(0.50.866i)T2 1 + (0.5 - 0.866i)T^{2}
79 1+(0.50.866i)T2 1 + (-0.5 - 0.866i)T^{2}
83 1+T+T2 1 + T + T^{2}
89 1+(0.50.866i)T2 1 + (-0.5 - 0.866i)T^{2}
97 1+T2 1 + T^{2}
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   L(s)=p j=12(1αj,pps)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

−8.659330615228117947383402275993, −8.470507690608842951038007020025, −7.57033008933390230494106598631, −7.30165757746488259783406120048, −5.62415852236648704299354914853, −5.05820642195303694774687866612, −4.37513906216833030932583372844, −3.46851343277349167427868613284, −2.21075619522109966132077998261, −0.72206824961062071798645446657, 1.35089012953080285445316301409, 2.95540018165320772393406118273, 3.69695136520859988420538150719, 4.27568027461664598340555195316, 5.61091398054100387808665124673, 6.22611235115667258567388439865, 7.40471070474306594963870286449, 7.911992987174439869646672830492, 8.246857951311914064867923837188, 9.447518534286920168368871786696

Graph of the ZZ-function along the critical line