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Properties

Label 2-275-5.4-c3-0-35
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  − 1.43i·2-s − 0.561i·3-s + 5.93·4-s − 0.807·6-s − 31.0i·7-s − 20.0i·8-s + 26.6·9-s − 11·11-s − 3.33i·12-s + 45.6i·13-s − 44.6·14-s + 18.6·16-s − 40.4i·17-s − 38.3i·18-s − 91.2·19-s + ⋯
L(s)  = 1  − 0.508i·2-s − 0.108i·3-s + 0.741·4-s − 0.0549·6-s − 1.67i·7-s − 0.885i·8-s + 0.988·9-s − 0.301·11-s − 0.0801i·12-s + 0.973i·13-s − 0.852·14-s + 0.290·16-s − 0.577i·17-s − 0.502i·18-s − 1.10·19-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 2.1802964652.180296465
L(12)L(\frac12) ≈\approx 2.1802964652.180296465
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+1.43iT−8T2 1 + 1.43iT - 8T^{2}
3 1+0.561iT−27T2 1 + 0.561iT - 27T^{2}
7 1+31.0iT−343T2 1 + 31.0iT - 343T^{2}
13 1−45.6iT−2.19e3T2 1 - 45.6iT - 2.19e3T^{2}
17 1+40.4iT−4.91e3T2 1 + 40.4iT - 4.91e3T^{2}
19 1+91.2T+6.85e3T2 1 + 91.2T + 6.85e3T^{2}
23 1+32.2iT−1.21e4T2 1 + 32.2iT - 1.21e4T^{2}
29 1+35.8T+2.43e4T2 1 + 35.8T + 2.43e4T^{2}
31 1−311.T+2.97e4T2 1 - 311.T + 2.97e4T^{2}
37 1+368.iT−5.06e4T2 1 + 368. iT - 5.06e4T^{2}
41 1+393.T+6.89e4T2 1 + 393.T + 6.89e4T^{2}
43 1−351.iT−7.95e4T2 1 - 351. iT - 7.95e4T^{2}
47 1+230.iT−1.03e5T2 1 + 230. iT - 1.03e5T^{2}
53 1+406.iT−1.48e5T2 1 + 406. iT - 1.48e5T^{2}
59 1−368.T+2.05e5T2 1 - 368.T + 2.05e5T^{2}
61 1+322.T+2.26e5T2 1 + 322.T + 2.26e5T^{2}
67 1−442.iT−3.00e5T2 1 - 442. iT - 3.00e5T^{2}
71 1−667.T+3.57e5T2 1 - 667.T + 3.57e5T^{2}
73 1−84.5iT−3.89e5T2 1 - 84.5iT - 3.89e5T^{2}
79 1−411.T+4.93e5T2 1 - 411.T + 4.93e5T^{2}
83 1−835.iT−5.71e5T2 1 - 835. iT - 5.71e5T^{2}
89 1−799.T+7.04e5T2 1 - 799.T + 7.04e5T^{2}
97 1−768.iT−9.12e5T2 1 - 768. iT - 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

−11.02702119689305835611705154375, −10.34223296999676054283750595817, −9.673303994914895848570034510505, −8.027931237161871424392257441560, −6.98980004613168497416407331010, −6.61284201340044846671770821204, −4.56920389883350203371250224072, −3.73225685174482483524338269278, −2.09739132238555092272592116348, −0.832830105290406522855055924617, 1.87125566795404075754337128300, 3.00890095887306138394614572558, 4.87008820175990593003947564022, 5.88275957735464605953974101622, 6.66123507156817981216777913513, 7.980775333456928087718060525912, 8.583095001214185576224583760089, 9.951403748914494922225129515742, 10.74527572939234908913562845972, 11.94794193790170820643664275543

Graph of the ZZ-function along the critical line