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Properties

Label 2-273-13.12-c1-0-6
Degree 22
Conductor 273273
Sign 0.911+0.410i0.911 + 0.410i
Analytic cond. 2.179912.17991
Root an. cond. 1.476451.47645
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  − 1.48i·2-s + 3-s − 0.193·4-s + 4.15i·5-s − 1.48i·6-s + i·7-s − 2.67i·8-s + 9-s + 6.15·10-s + 3.19i·11-s − 0.193·12-s + (1.48 − 3.28i)13-s + 1.48·14-s + 4.15i·15-s − 4.35·16-s + 3.35·17-s + ⋯
L(s)  = 1  − 1.04i·2-s + 0.577·3-s − 0.0969·4-s + 1.85i·5-s − 0.604i·6-s + 0.377i·7-s − 0.945i·8-s + 0.333·9-s + 1.94·10-s + 0.963i·11-s − 0.0559·12-s + (0.410 − 0.911i)13-s + 0.395·14-s + 1.07i·15-s − 1.08·16-s + 0.812·17-s + ⋯

Functional equation

Λ(s)=(273s/2 ΓC(s) L(s)=((0.911+0.410i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(273s/2 ΓC(s+1/2) L(s)=((0.911+0.410i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 273273    =    3⋅7⋅133 \cdot 7 \cdot 13
Sign: 0.911+0.410i0.911 + 0.410i
Analytic conductor: 2.179912.17991
Root analytic conductor: 1.476451.47645
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ273(64,⋅)\chi_{273} (64, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 273, ( :1/2), 0.911+0.410i)(2,\ 273,\ (\ :1/2),\ 0.911 + 0.410i)

Particular Values

L(1)L(1) ≈\approx 1.62783−0.349804i1.62783 - 0.349804i
L(12)L(\frac12) ≈\approx 1.62783−0.349804i1.62783 - 0.349804i
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)
bad3 1−T 1 - T
7 1−iT 1 - iT
13 1+(−1.48+3.28i)T 1 + (-1.48 + 3.28i)T
good2 1+1.48iT−2T2 1 + 1.48iT - 2T^{2}
5 1−4.15iT−5T2 1 - 4.15iT - 5T^{2}
11 1−3.19iT−11T2 1 - 3.19iT - 11T^{2}
17 1−3.35T+17T2 1 - 3.35T + 17T^{2}
19 1+2.38iT−19T2 1 + 2.38iT - 19T^{2}
23 1−0.387T+23T2 1 - 0.387T + 23T^{2}
29 1+7.92T+29T2 1 + 7.92T + 29T^{2}
31 1+10.7iT−31T2 1 + 10.7iT - 31T^{2}
37 1−1.61iT−37T2 1 - 1.61iT - 37T^{2}
41 1+1.45iT−41T2 1 + 1.45iT - 41T^{2}
43 1−1.92T+43T2 1 - 1.92T + 43T^{2}
47 1−3.76iT−47T2 1 - 3.76iT - 47T^{2}
53 1+6T+53T2 1 + 6T + 53T^{2}
59 1−6.15iT−59T2 1 - 6.15iT - 59T^{2}
61 1−14.4T+61T2 1 - 14.4T + 61T^{2}
67 1+5.61iT−67T2 1 + 5.61iT - 67T^{2}
71 1+11.8iT−71T2 1 + 11.8iT - 71T^{2}
73 1−15.6iT−73T2 1 - 15.6iT - 73T^{2}
79 1+8.96T+79T2 1 + 8.96T + 79T^{2}
83 1+6.99iT−83T2 1 + 6.99iT - 83T^{2}
89 1+0.932iT−89T2 1 + 0.932iT - 89T^{2}
97 1−3.35iT−97T2 1 - 3.35iT - 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

−11.56720476208974907813891072134, −10.94397564747974915950440206639, −10.06455916659142812126974281203, −9.516988883160988772998483656947, −7.77011228778174988486872789695, −7.09291363604466615922813667379, −5.94391882187187924690171256339, −3.87614953088857811018814751358, −2.96676757074889381464756178226, −2.14084616359296801334897363508, 1.52575099620419627149295276262, 3.76025364319328962582934486898, 5.03503493557321978763522096659, 5.88370679230778932494542565513, 7.19932835911843331415049322696, 8.273983413923977410220968786711, 8.662685200341064569525488039054, 9.623737152634746465591811302880, 11.13230041507178718147523479814, 12.13448636713365840681108619974

Graph of the ZZ-function along the critical line