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Properties

Label 2-1127-161.51-c0-0-0
Degree 22
Conductor 11271127
Sign −0.822+0.568i-0.822 + 0.568i
Analytic cond. 0.5624460.562446
Root an. cond. 0.7499640.749964
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  + (−1.21 + 1.16i)2-s + (0.0871 − 1.82i)4-s + (0.915 + 1.05i)8-s + (−0.928 − 0.371i)9-s + (−1.04 + 1.09i)11-s + (−0.518 − 0.0495i)16-s + (1.56 − 0.625i)18-s − 2.54i·22-s + (−0.786 − 0.618i)23-s + (0.235 + 0.971i)25-s + (−1.61 + 1.03i)29-s + (−0.409 + 0.322i)32-s + (−0.760 + 1.66i)36-s + (−0.676 + 1.68i)37-s + (−1.49 − 1.29i)43-s + (1.90 + 2.00i)44-s + ⋯
L(s)  = 1  + (−1.21 + 1.16i)2-s + (0.0871 − 1.82i)4-s + (0.915 + 1.05i)8-s + (−0.928 − 0.371i)9-s + (−1.04 + 1.09i)11-s + (−0.518 − 0.0495i)16-s + (1.56 − 0.625i)18-s − 2.54i·22-s + (−0.786 − 0.618i)23-s + (0.235 + 0.971i)25-s + (−1.61 + 1.03i)29-s + (−0.409 + 0.322i)32-s + (−0.760 + 1.66i)36-s + (−0.676 + 1.68i)37-s + (−1.49 − 1.29i)43-s + (1.90 + 2.00i)44-s + ⋯

Functional equation

Λ(s)=(1127s/2 ΓC(s) L(s)=((−0.822+0.568i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1127s/2 ΓC(s) L(s)=((−0.822+0.568i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11271127    =    72⋅237^{2} \cdot 23
Sign: −0.822+0.568i-0.822 + 0.568i
Analytic conductor: 0.5624460.562446
Root analytic conductor: 0.7499640.749964
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1127(373,⋅)\chi_{1127} (373, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1127, ( :0), −0.822+0.568i)(2,\ 1127,\ (\ :0),\ -0.822 + 0.568i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.15880610680.1588061068
L(12)L(\frac12) ≈\approx 0.15880610680.1588061068
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)
bad7 1 1
23 1+(0.786+0.618i)T 1 + (0.786 + 0.618i)T
good2 1+(1.21−1.16i)T+(0.0475−0.998i)T2 1 + (1.21 - 1.16i)T + (0.0475 - 0.998i)T^{2}
3 1+(0.928+0.371i)T2 1 + (0.928 + 0.371i)T^{2}
5 1+(−0.235−0.971i)T2 1 + (-0.235 - 0.971i)T^{2}
11 1+(1.04−1.09i)T+(−0.0475−0.998i)T2 1 + (1.04 - 1.09i)T + (-0.0475 - 0.998i)T^{2}
13 1+(−0.654−0.755i)T2 1 + (-0.654 - 0.755i)T^{2}
17 1+(−0.580−0.814i)T2 1 + (-0.580 - 0.814i)T^{2}
19 1+(−0.580+0.814i)T2 1 + (-0.580 + 0.814i)T^{2}
29 1+(1.61−1.03i)T+(0.415−0.909i)T2 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2}
31 1+(−0.786−0.618i)T2 1 + (-0.786 - 0.618i)T^{2}
37 1+(0.676−1.68i)T+(−0.723−0.690i)T2 1 + (0.676 - 1.68i)T + (-0.723 - 0.690i)T^{2}
41 1+(−0.959−0.281i)T2 1 + (-0.959 - 0.281i)T^{2}
43 1+(1.49+1.29i)T+(0.142+0.989i)T2 1 + (1.49 + 1.29i)T + (0.142 + 0.989i)T^{2}
47 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
53 1+(0.458+0.326i)T+(0.327+0.945i)T2 1 + (0.458 + 0.326i)T + (0.327 + 0.945i)T^{2}
59 1+(0.981−0.189i)T2 1 + (0.981 - 0.189i)T^{2}
61 1+(−0.928+0.371i)T2 1 + (-0.928 + 0.371i)T^{2}
67 1+(1.05−0.254i)T+(0.888−0.458i)T2 1 + (1.05 - 0.254i)T + (0.888 - 0.458i)T^{2}
71 1+(−1.25−0.368i)T+(0.841+0.540i)T2 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2}
73 1+(−0.995−0.0950i)T2 1 + (-0.995 - 0.0950i)T^{2}
79 1+(−0.458+0.326i)T+(0.327−0.945i)T2 1 + (-0.458 + 0.326i)T + (0.327 - 0.945i)T^{2}
83 1+(0.959−0.281i)T2 1 + (0.959 - 0.281i)T^{2}
89 1+(0.786−0.618i)T2 1 + (0.786 - 0.618i)T^{2}
97 1+(0.959+0.281i)T2 1 + (0.959 + 0.281i)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

−10.18417018517938085996202522987, −9.522588321694323021800064707849, −8.711170869440704057100578561463, −8.103400461057385336068831969167, −7.27468684056163305824569445437, −6.64876071869805201424143201850, −5.60839983286456532273980192027, −5.00055178782366461070820043954, −3.34283693728567222645028487550, −1.84554381322945571023078866229, 0.19768603830847262584629379663, 1.99346829424782372295086047028, 2.85584533417621086714048230576, 3.74790161829736804324663354684, 5.30840971467327790311750213119, 6.10602202932046425481141835455, 7.70013880340093872336832478224, 8.059073568092085010186376372206, 8.831199150534231085891276733211, 9.594318539804705250644773392227

Graph of the ZZ-function along the critical line