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Properties

Label 2-253-11.10-c2-0-29
Degree 22
Conductor 253253
Sign 0.808+0.588i0.808 + 0.588i
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  + 1.81i·2-s − 2.41·3-s + 0.713·4-s + 4.05·5-s − 4.37i·6-s − 13.5i·7-s + 8.54i·8-s − 3.17·9-s + 7.34i·10-s + (−8.89 − 6.47i)11-s − 1.72·12-s − 22.1i·13-s + 24.4·14-s − 9.78·15-s − 12.6·16-s + 2.51i·17-s + ⋯
L(s)  = 1  + 0.906i·2-s − 0.804·3-s + 0.178·4-s + 0.810·5-s − 0.729i·6-s − 1.92i·7-s + 1.06i·8-s − 0.352·9-s + 0.734i·10-s + (−0.808 − 0.588i)11-s − 0.143·12-s − 1.70i·13-s + 1.74·14-s − 0.652·15-s − 0.789·16-s + 0.148i·17-s + ⋯

Functional equation

Λ(s)=(253s/2 ΓC(s) L(s)=((0.808+0.588i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(253s/2 ΓC(s+1) L(s)=((0.808+0.588i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 253253    =    11⋅2311 \cdot 23
Sign: 0.808+0.588i0.808 + 0.588i
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), 0.808+0.588i)(2,\ 253,\ (\ :1),\ 0.808 + 0.588i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.15187−0.374805i1.15187 - 0.374805i
L(12)L(\frac12) ≈\approx 1.15187−0.374805i1.15187 - 0.374805i
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+(8.89+6.47i)T 1 + (8.89 + 6.47i)T
23 1+4.79T 1 + 4.79T
good2 1−1.81iT−4T2 1 - 1.81iT - 4T^{2}
3 1+2.41T+9T2 1 + 2.41T + 9T^{2}
5 1−4.05T+25T2 1 - 4.05T + 25T^{2}
7 1+13.5iT−49T2 1 + 13.5iT - 49T^{2}
13 1+22.1iT−169T2 1 + 22.1iT - 169T^{2}
17 1−2.51iT−289T2 1 - 2.51iT - 289T^{2}
19 1+18.9iT−361T2 1 + 18.9iT - 361T^{2}
29 1−44.4iT−841T2 1 - 44.4iT - 841T^{2}
31 1−58.9T+961T2 1 - 58.9T + 961T^{2}
37 1−27.8T+1.36e3T2 1 - 27.8T + 1.36e3T^{2}
41 1+35.5iT−1.68e3T2 1 + 35.5iT - 1.68e3T^{2}
43 1+23.8iT−1.84e3T2 1 + 23.8iT - 1.84e3T^{2}
47 1+18.3T+2.20e3T2 1 + 18.3T + 2.20e3T^{2}
53 1+10.6T+2.80e3T2 1 + 10.6T + 2.80e3T^{2}
59 1−12.3T+3.48e3T2 1 - 12.3T + 3.48e3T^{2}
61 1−33.3iT−3.72e3T2 1 - 33.3iT - 3.72e3T^{2}
67 1−97.8T+4.48e3T2 1 - 97.8T + 4.48e3T^{2}
71 1+92.8T+5.04e3T2 1 + 92.8T + 5.04e3T^{2}
73 1−59.0iT−5.32e3T2 1 - 59.0iT - 5.32e3T^{2}
79 1+82.0iT−6.24e3T2 1 + 82.0iT - 6.24e3T^{2}
83 1−26.5iT−6.88e3T2 1 - 26.5iT - 6.88e3T^{2}
89 1−28.0T+7.92e3T2 1 - 28.0T + 7.92e3T^{2}
97 1−54.0T+9.40e3T2 1 - 54.0T + 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

−11.42822259386098573082594804948, −10.59842350051955697717219236403, −10.25868143392952325126766686115, −8.416312285675155369432580693553, −7.56123355774180777346553048057, −6.61041044618905311783284873914, −5.72389038151155898348591884794, −4.92952150009889541707984818932, −2.96613948138804081514354623760, −0.66297517991029214456001012008, 1.93971096835823087078090457115, 2.64647919345810754359469459312, 4.68927096150755117192681309527, 5.97753023333632246730555972250, 6.36052728904400159196611695929, 8.199477937702869170533707662943, 9.521229274946354392296020159221, 9.941117692062691845218146661341, 11.33648444666270011088424814178, 11.81220817636649018582114065116

Graph of the ZZ-function along the critical line