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Properties

Label 2-5220-145.144-c1-0-1
Degree 22
Conductor 52205220
Sign −0.794+0.607i-0.794 + 0.607i
Analytic cond. 41.681941.6819
Root an. cond. 6.456156.45615
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.591 + 2.15i)5-s + 2.69i·7-s − 0.968i·11-s − 2.99i·13-s − 1.02·17-s + 7.89i·19-s − 0.0832i·23-s + (−4.29 − 2.55i)25-s + (−4.28 − 3.25i)29-s + 2.02i·31-s + (−5.81 − 1.59i)35-s + 3.06·37-s + 3.92i·41-s − 7.15·43-s + 0.554·47-s + ⋯
L(s)  = 1  + (−0.264 + 0.964i)5-s + 1.01i·7-s − 0.291i·11-s − 0.831i·13-s − 0.249·17-s + 1.81i·19-s − 0.0173i·23-s + (−0.859 − 0.510i)25-s + (−0.796 − 0.604i)29-s + 0.363i·31-s + (−0.983 − 0.269i)35-s + 0.503·37-s + 0.613i·41-s − 1.09·43-s + 0.0808·47-s + ⋯

Functional equation

Λ(s)=(5220s/2 ΓC(s) L(s)=((−0.794+0.607i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(5220s/2 ΓC(s+1/2) L(s)=((−0.794+0.607i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 52205220    =    22⋅32⋅5⋅292^{2} \cdot 3^{2} \cdot 5 \cdot 29
Sign: −0.794+0.607i-0.794 + 0.607i
Analytic conductor: 41.681941.6819
Root analytic conductor: 6.456156.45615
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ5220(289,⋅)\chi_{5220} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 5220, ( :1/2), −0.794+0.607i)(2,\ 5220,\ (\ :1/2),\ -0.794 + 0.607i)

Particular Values

L(1)L(1) ≈\approx 0.33598837190.3359883719
L(12)L(\frac12) ≈\approx 0.33598837190.3359883719
L(32)L(\frac{3}{2}) 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
3 1 1
5 1+(0.591−2.15i)T 1 + (0.591 - 2.15i)T
29 1+(4.28+3.25i)T 1 + (4.28 + 3.25i)T
good7 1−2.69iT−7T2 1 - 2.69iT - 7T^{2}
11 1+0.968iT−11T2 1 + 0.968iT - 11T^{2}
13 1+2.99iT−13T2 1 + 2.99iT - 13T^{2}
17 1+1.02T+17T2 1 + 1.02T + 17T^{2}
19 1−7.89iT−19T2 1 - 7.89iT - 19T^{2}
23 1+0.0832iT−23T2 1 + 0.0832iT - 23T^{2}
31 1−2.02iT−31T2 1 - 2.02iT - 31T^{2}
37 1−3.06T+37T2 1 - 3.06T + 37T^{2}
41 1−3.92iT−41T2 1 - 3.92iT - 41T^{2}
43 1+7.15T+43T2 1 + 7.15T + 43T^{2}
47 1−0.554T+47T2 1 - 0.554T + 47T^{2}
53 1+6.25iT−53T2 1 + 6.25iT - 53T^{2}
59 1+11.5T+59T2 1 + 11.5T + 59T^{2}
61 1−8.77iT−61T2 1 - 8.77iT - 61T^{2}
67 1+2.40iT−67T2 1 + 2.40iT - 67T^{2}
71 1+11.1T+71T2 1 + 11.1T + 71T^{2}
73 1−7.48T+73T2 1 - 7.48T + 73T^{2}
79 1−6.55iT−79T2 1 - 6.55iT - 79T^{2}
83 1+7.10iT−83T2 1 + 7.10iT - 83T^{2}
89 1+6.83iT−89T2 1 + 6.83iT - 89T^{2}
97 1−4.18T+97T2 1 - 4.18T + 97T^{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

−8.440866519984943571054662768387, −7.963318498552394825206605854138, −7.31213683032015144815176885430, −6.25294137800130840337624848175, −5.94704282473884271667856972421, −5.15934334347246533837046599688, −4.01703583208055843863404906977, −3.29322570919226375473968722690, −2.59522326269529533449130679488, −1.63683266822024441043464242315, 0.092051216766765740728830424629, 1.14364805732313749406575629008, 2.12989143859381292141297498115, 3.34857548113760011289539615750, 4.28194473839164463349335149042, 4.62430559079127763781271749176, 5.43309345957566896229503914113, 6.50335600873075520083524070104, 7.13694393638585073384234710401, 7.67178159834842219630330756967

Graph of the ZZ-function along the critical line