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Properties

Label 2-275-5.4-c3-0-14
Degree 22
Conductor 275275
Sign −0.447−0.894i-0.447 - 0.894i
Analytic cond. 16.225516.2255
Root an. cond. 4.028094.02809
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.56i·2-s − 3.56i·3-s − 22.9·4-s + 19.8·6-s − 6.05i·7-s − 83.0i·8-s + 14.3·9-s − 11·11-s + 81.6i·12-s − 4.38i·13-s + 33.6·14-s + 278.·16-s + 110. i·17-s + 79.6i·18-s + 94.2·19-s + ⋯
L(s)  = 1  + 1.96i·2-s − 0.685i·3-s − 2.86·4-s + 1.34·6-s − 0.326i·7-s − 3.66i·8-s + 0.530·9-s − 0.301·11-s + 1.96i·12-s − 0.0935i·13-s + 0.642·14-s + 4.34·16-s + 1.57i·17-s + 1.04i·18-s + 1.13·19-s + ⋯

Functional equation

Λ(s)=(275s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(275s/2 ΓC(s+3/2) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 275275    =    52⋅115^{2} \cdot 11
Sign: −0.447−0.894i-0.447 - 0.894i
Analytic conductor: 16.225516.2255
Root analytic conductor: 4.028094.02809
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ275(199,⋅)\chi_{275} (199, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 275, ( :3/2), −0.447−0.894i)(2,\ 275,\ (\ :3/2),\ -0.447 - 0.894i)

Particular Values

L(2)L(2) ≈\approx 1.4691598861.469159886
L(12)L(\frac12) ≈\approx 1.4691598861.469159886
L(52)L(\frac{5}{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)
bad5 1 1
11 1+11T 1 + 11T
good2 1−5.56iT−8T2 1 - 5.56iT - 8T^{2}
3 1+3.56iT−27T2 1 + 3.56iT - 27T^{2}
7 1+6.05iT−343T2 1 + 6.05iT - 343T^{2}
13 1+4.38iT−2.19e3T2 1 + 4.38iT - 2.19e3T^{2}
17 1−110.iT−4.91e3T2 1 - 110. iT - 4.91e3T^{2}
19 1−94.2T+6.85e3T2 1 - 94.2T + 6.85e3T^{2}
23 1−15.7iT−1.21e4T2 1 - 15.7iT - 1.21e4T^{2}
29 1−256.T+2.43e4T2 1 - 256.T + 2.43e4T^{2}
31 1+170.T+2.97e4T2 1 + 170.T + 2.97e4T^{2}
37 1−190.iT−5.06e4T2 1 - 190. iT - 5.06e4T^{2}
41 1−249.T+6.89e4T2 1 - 249.T + 6.89e4T^{2}
43 1−291.iT−7.95e4T2 1 - 291. iT - 7.95e4T^{2}
47 1+182.iT−1.03e5T2 1 + 182. iT - 1.03e5T^{2}
53 1+289.iT−1.48e5T2 1 + 289. iT - 1.48e5T^{2}
59 1+282.T+2.05e5T2 1 + 282.T + 2.05e5T^{2}
61 1−167.T+2.26e5T2 1 - 167.T + 2.26e5T^{2}
67 1−176.iT−3.00e5T2 1 - 176. iT - 3.00e5T^{2}
71 1−919.T+3.57e5T2 1 - 919.T + 3.57e5T^{2}
73 1−154.iT−3.89e5T2 1 - 154. iT - 3.89e5T^{2}
79 1−882.T+4.93e5T2 1 - 882.T + 4.93e5T^{2}
83 1−277.iT−5.71e5T2 1 - 277. iT - 5.71e5T^{2}
89 1−977.T+7.04e5T2 1 - 977.T + 7.04e5T^{2}
97 1−1.10e3iT−9.12e5T2 1 - 1.10e3iT - 9.12e5T^{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

−12.30365211915230958156200862246, −10.41760033954673995833860973428, −9.536114399029200935513739095797, −8.316047063313988640208429015931, −7.73216574883631815875090944214, −6.85581944849055318227905035804, −6.08810398602743430818688444520, −4.96016855131287772733825271484, −3.80884362878297945976619025775, −1.00029397773553886878634320796, 0.818520054576956611350462280492, 2.41681227933541063152020160258, 3.46382616359390149130769786299, 4.60183254571979644935984580149, 5.34444877455118688922572436657, 7.56588585431064936082739177274, 9.008953571912181824933498590885, 9.475432262649928472834336641997, 10.33821796048451263676251227891, 11.06684119445269040153759689065

Graph of the ZZ-function along the critical line