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Properties

Label 2-253-11.10-c2-0-33
Degree 22
Conductor 253253
Sign 0.831+0.555i0.831 + 0.555i
Analytic cond. 6.893756.89375
Root an. cond. 2.625592.62559
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.919i·2-s + 4.75·3-s + 3.15·4-s + 0.241·5-s − 4.37i·6-s + 3.10i·7-s − 6.58i·8-s + 13.5·9-s − 0.222i·10-s + (−9.14 − 6.11i)11-s + 14.9·12-s + 4.95i·13-s + 2.85·14-s + 1.14·15-s + 6.56·16-s − 23.5i·17-s + ⋯
L(s)  = 1  − 0.459i·2-s + 1.58·3-s + 0.788·4-s + 0.0482·5-s − 0.728i·6-s + 0.442i·7-s − 0.822i·8-s + 1.51·9-s − 0.0222i·10-s + (−0.831 − 0.555i)11-s + 1.24·12-s + 0.381i·13-s + 0.203·14-s + 0.0764·15-s + 0.410·16-s − 1.38i·17-s + ⋯

Functional equation

Λ(s)=(253s/2 ΓC(s) L(s)=((0.831+0.555i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(253s/2 ΓC(s+1) L(s)=((0.831+0.555i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 253253    =    11⋅2311 \cdot 23
Sign: 0.831+0.555i0.831 + 0.555i
Analytic conductor: 6.893756.89375
Root analytic conductor: 2.625592.62559
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ253(208,⋅)\chi_{253} (208, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 253, ( :1), 0.831+0.555i)(2,\ 253,\ (\ :1),\ 0.831 + 0.555i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.89515−0.878949i2.89515 - 0.878949i
L(12)L(\frac12) ≈\approx 2.89515−0.878949i2.89515 - 0.878949i
L(2)L(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)
bad11 1+(9.14+6.11i)T 1 + (9.14 + 6.11i)T
23 1+4.79T 1 + 4.79T
good2 1+0.919iT−4T2 1 + 0.919iT - 4T^{2}
3 1−4.75T+9T2 1 - 4.75T + 9T^{2}
5 1−0.241T+25T2 1 - 0.241T + 25T^{2}
7 1−3.10iT−49T2 1 - 3.10iT - 49T^{2}
13 1−4.95iT−169T2 1 - 4.95iT - 169T^{2}
17 1+23.5iT−289T2 1 + 23.5iT - 289T^{2}
19 1−32.7iT−361T2 1 - 32.7iT - 361T^{2}
29 1−47.1iT−841T2 1 - 47.1iT - 841T^{2}
31 1−25.6T+961T2 1 - 25.6T + 961T^{2}
37 1+50.3T+1.36e3T2 1 + 50.3T + 1.36e3T^{2}
41 1+13.0iT−1.68e3T2 1 + 13.0iT - 1.68e3T^{2}
43 1+48.3iT−1.84e3T2 1 + 48.3iT - 1.84e3T^{2}
47 1−30.1T+2.20e3T2 1 - 30.1T + 2.20e3T^{2}
53 1−5.53T+2.80e3T2 1 - 5.53T + 2.80e3T^{2}
59 1+32.5T+3.48e3T2 1 + 32.5T + 3.48e3T^{2}
61 1+83.2iT−3.72e3T2 1 + 83.2iT - 3.72e3T^{2}
67 1+127.T+4.48e3T2 1 + 127.T + 4.48e3T^{2}
71 1−40.3T+5.04e3T2 1 - 40.3T + 5.04e3T^{2}
73 1−57.4iT−5.32e3T2 1 - 57.4iT - 5.32e3T^{2}
79 1−76.1iT−6.24e3T2 1 - 76.1iT - 6.24e3T^{2}
83 1−115.iT−6.88e3T2 1 - 115. iT - 6.88e3T^{2}
89 1+46.9T+7.92e3T2 1 + 46.9T + 7.92e3T^{2}
97 1+14.0T+9.40e3T2 1 + 14.0T + 9.40e3T^{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

−11.87301785884365088882484604140, −10.60640669257089214097638060410, −9.809514222917220888643040508072, −8.815165906130787124844207509896, −7.906655268231652601396316851394, −7.04911086322654761955146013953, −5.57184635051785182892599675976, −3.70579577909634079438928489912, −2.83138906507665687286175078273, −1.83555093570280436255096303003, 2.03120355382228717685456172265, 2.97827659073528614468808206168, 4.40254305238262775673651999358, 6.03816831448492554399296574476, 7.30974081044183680662355442856, 7.86697355310214587785178507125, 8.708875117910097327353508478672, 9.950363325042522800774025239588, 10.69392785495282160385174966683, 11.99797043027612119785203755669

Graph of the ZZ-function along the critical line