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Properties

Label 2-15e2-5.3-c4-0-19
Degree 22
Conductor 225225
Sign 0.525+0.850i0.525 + 0.850i
Analytic cond. 23.258223.2582
Root an. cond. 4.822674.82267
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.30 − 3.30i)2-s − 5.84i·4-s + (33.1 − 33.1i)7-s + (33.5 + 33.5i)8-s − 55.3·11-s + (161. + 161. i)13-s − 219. i·14-s + 315.·16-s + (278. − 278. i)17-s + 179. i·19-s + (−182. + 182. i)22-s + (−398. − 398. i)23-s + 1.07e3·26-s + (−193. − 193. i)28-s − 547. i·29-s + ⋯
L(s)  = 1  + (0.826 − 0.826i)2-s − 0.365i·4-s + (0.676 − 0.676i)7-s + (0.524 + 0.524i)8-s − 0.457·11-s + (0.958 + 0.958i)13-s − 1.11i·14-s + 1.23·16-s + (0.965 − 0.965i)17-s + 0.498i·19-s + (−0.377 + 0.377i)22-s + (−0.752 − 0.752i)23-s + 1.58·26-s + (−0.247 − 0.247i)28-s − 0.650i·29-s + ⋯

Functional equation

Λ(s)=(225s/2 ΓC(s) L(s)=((0.525+0.850i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(225s/2 ΓC(s+2) L(s)=((0.525+0.850i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 225225    =    32⋅523^{2} \cdot 5^{2}
Sign: 0.525+0.850i0.525 + 0.850i
Analytic conductor: 23.258223.2582
Root analytic conductor: 4.822674.82267
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ225(118,⋅)\chi_{225} (118, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 225, ( :2), 0.525+0.850i)(2,\ 225,\ (\ :2),\ 0.525 + 0.850i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 3.5093449573.509344957
L(12)L(\frac12) ≈\approx 3.5093449573.509344957
L(3)L(3) 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 1
5 1 1
good2 1+(−3.30+3.30i)T−16iT2 1 + (-3.30 + 3.30i)T - 16iT^{2}
7 1+(−33.1+33.1i)T−2.40e3iT2 1 + (-33.1 + 33.1i)T - 2.40e3iT^{2}
11 1+55.3T+1.46e4T2 1 + 55.3T + 1.46e4T^{2}
13 1+(−161.−161.i)T+2.85e4iT2 1 + (-161. - 161. i)T + 2.85e4iT^{2}
17 1+(−278.+278.i)T−8.35e4iT2 1 + (-278. + 278. i)T - 8.35e4iT^{2}
19 1−179.iT−1.30e5T2 1 - 179. iT - 1.30e5T^{2}
23 1+(398.+398.i)T+2.79e5iT2 1 + (398. + 398. i)T + 2.79e5iT^{2}
29 1+547.iT−7.07e5T2 1 + 547. iT - 7.07e5T^{2}
31 1−1.53e3T+9.23e5T2 1 - 1.53e3T + 9.23e5T^{2}
37 1+(−1.66e3+1.66e3i)T−1.87e6iT2 1 + (-1.66e3 + 1.66e3i)T - 1.87e6iT^{2}
41 1−307.T+2.82e6T2 1 - 307.T + 2.82e6T^{2}
43 1+(104.+104.i)T+3.41e6iT2 1 + (104. + 104. i)T + 3.41e6iT^{2}
47 1+(346.−346.i)T−4.87e6iT2 1 + (346. - 346. i)T - 4.87e6iT^{2}
53 1+(2.02e3+2.02e3i)T+7.89e6iT2 1 + (2.02e3 + 2.02e3i)T + 7.89e6iT^{2}
59 1−2.85e3iT−1.21e7T2 1 - 2.85e3iT - 1.21e7T^{2}
61 1+1.05e3T+1.38e7T2 1 + 1.05e3T + 1.38e7T^{2}
67 1+(3.75e3−3.75e3i)T−2.01e7iT2 1 + (3.75e3 - 3.75e3i)T - 2.01e7iT^{2}
71 1+1.42e3T+2.54e7T2 1 + 1.42e3T + 2.54e7T^{2}
73 1+(813.+813.i)T+2.83e7iT2 1 + (813. + 813. i)T + 2.83e7iT^{2}
79 1−4.85e3iT−3.89e7T2 1 - 4.85e3iT - 3.89e7T^{2}
83 1+(1.31e3+1.31e3i)T+4.74e7iT2 1 + (1.31e3 + 1.31e3i)T + 4.74e7iT^{2}
89 1−5.18e3iT−6.27e7T2 1 - 5.18e3iT - 6.27e7T^{2}
97 1+(3.49e3−3.49e3i)T−8.85e7iT2 1 + (3.49e3 - 3.49e3i)T - 8.85e7iT^{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

−11.54732851941957759558373224133, −10.78161526033274485183844726136, −9.828357063166986677878408926373, −8.310824417953207435416675563812, −7.53349567675609078341672617129, −6.00321153879035799589550551748, −4.67938008087282857405080999282, −3.93239174201237433102766955406, −2.55883133983498877921029442982, −1.17598293090456730018618480934, 1.31507940831095281010337193121, 3.22662499145853306147369668345, 4.62428100242563282904142318084, 5.61461119493455474941683672478, 6.26149694854711837224983402248, 7.76991249108865868247502742261, 8.335194081494378176646535214084, 9.894354372083985546191315586636, 10.79054868696523020745866168907, 11.95545548300864286612832235071

Graph of the ZZ-function along the critical line