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Properties

Label 2-2793-399.356-c0-0-0
Degree 22
Conductor 27932793
Sign 0.671−0.740i0.671 - 0.740i
Analytic cond. 1.393881.39388
Root an. cond. 1.180631.18063
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  + (0.173 − 0.984i)3-s + (−0.766 + 0.642i)4-s + (−0.939 − 0.342i)9-s + (0.5 + 0.866i)12-s + (0.326 + 1.85i)13-s + (0.173 − 0.984i)16-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)25-s + (−0.5 + 0.866i)27-s + (0.766 + 1.32i)31-s + (0.939 − 0.342i)36-s + 1.96i·37-s + 1.87·39-s + (−0.266 − 0.223i)43-s + (−0.939 − 0.342i)48-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)3-s + (−0.766 + 0.642i)4-s + (−0.939 − 0.342i)9-s + (0.5 + 0.866i)12-s + (0.326 + 1.85i)13-s + (0.173 − 0.984i)16-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)25-s + (−0.5 + 0.866i)27-s + (0.766 + 1.32i)31-s + (0.939 − 0.342i)36-s + 1.96i·37-s + 1.87·39-s + (−0.266 − 0.223i)43-s + (−0.939 − 0.342i)48-s + ⋯

Functional equation

Λ(s)=(2793s/2 ΓC(s) L(s)=((0.671−0.740i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2793s/2 ΓC(s) L(s)=((0.671−0.740i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27932793    =    3⋅72⋅193 \cdot 7^{2} \cdot 19
Sign: 0.671−0.740i0.671 - 0.740i
Analytic conductor: 1.393881.39388
Root analytic conductor: 1.180631.18063
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2793(2351,⋅)\chi_{2793} (2351, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2793, ( :0), 0.671−0.740i)(2,\ 2793,\ (\ :0),\ 0.671 - 0.740i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.87564664450.8756466445
L(12)L(\frac12) ≈\approx 0.87564664450.8756466445
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)
bad3 1+(−0.173+0.984i)T 1 + (-0.173 + 0.984i)T
7 1 1
19 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good2 1+(0.766−0.642i)T2 1 + (0.766 - 0.642i)T^{2}
5 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
13 1+(−0.326−1.85i)T+(−0.939+0.342i)T2 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2}
17 1+(0.766−0.642i)T2 1 + (0.766 - 0.642i)T^{2}
23 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
29 1+(0.766+0.642i)T2 1 + (0.766 + 0.642i)T^{2}
31 1+(−0.766−1.32i)T+(−0.5+0.866i)T2 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2}
37 1−1.96iT−T2 1 - 1.96iT - T^{2}
41 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
43 1+(0.266+0.223i)T+(0.173+0.984i)T2 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2}
47 1+(0.766+0.642i)T2 1 + (0.766 + 0.642i)T^{2}
53 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)T^{2}
59 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
61 1+(−0.826−0.984i)T+(−0.173+0.984i)T2 1 + (-0.826 - 0.984i)T + (-0.173 + 0.984i)T^{2}
67 1+(−0.233+0.642i)T+(−0.766−0.642i)T2 1 + (-0.233 + 0.642i)T + (-0.766 - 0.642i)T^{2}
71 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
73 1+(−0.673−0.118i)T+(0.939+0.342i)T2 1 + (-0.673 - 0.118i)T + (0.939 + 0.342i)T^{2}
79 1+(−1.26−0.223i)T+(0.939+0.342i)T2 1 + (-1.26 - 0.223i)T + (0.939 + 0.342i)T^{2}
83 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
89 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
97 1+(−0.939+0.342i)T+(0.766−0.642i)T2 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)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

−8.789108078231855338346435219014, −8.395433708400424953023200059000, −7.72211384550965562831810656361, −6.67309757852154338332769288653, −6.42560153442304587196720201755, −5.13803011437355749258103386355, −4.28634591160026558106083749072, −3.48613729844348860055313343403, −2.42371076888873624515443201298, −1.35074426434797745995099839904, 0.59781308088336191949865761881, 2.36170627648345842047261750464, 3.44040312021017779341716224720, 4.14598495543983918908964801258, 5.09620598327130620208992632977, 5.54193307198223537649088243892, 6.30179447144839747937629523180, 7.66258762209651927777942920191, 8.295290488430041898843808910311, 9.034820468552344851215424127829

Graph of the ZZ-function along the critical line