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Properties

Label 2-253-11.10-c2-0-7
Degree 22
Conductor 253253
Sign −0.856+0.516i-0.856 + 0.516i
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  + 3.08i·2-s − 3.11·3-s − 5.53·4-s + 8.57·5-s − 9.62i·6-s + 4.55i·7-s − 4.72i·8-s + 0.725·9-s + 26.4i·10-s + (−9.41 + 5.68i)11-s + 17.2·12-s + 5.40i·13-s − 14.0·14-s − 26.7·15-s − 7.52·16-s + 30.3i·17-s + ⋯
L(s)  = 1  + 1.54i·2-s − 1.03·3-s − 1.38·4-s + 1.71·5-s − 1.60i·6-s + 0.650i·7-s − 0.591i·8-s + 0.0805·9-s + 2.64i·10-s + (−0.856 + 0.516i)11-s + 1.43·12-s + 0.415i·13-s − 1.00·14-s − 1.78·15-s − 0.470·16-s + 1.78i·17-s + ⋯

Functional equation

Λ(s)=(253s/2 ΓC(s) L(s)=((−0.856+0.516i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(253s/2 ΓC(s+1) L(s)=((−0.856+0.516i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 253253    =    11⋅2311 \cdot 23
Sign: −0.856+0.516i-0.856 + 0.516i
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), −0.856+0.516i)(2,\ 253,\ (\ :1),\ -0.856 + 0.516i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.269671−0.968965i0.269671 - 0.968965i
L(12)L(\frac12) ≈\approx 0.269671−0.968965i0.269671 - 0.968965i
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+(9.41−5.68i)T 1 + (9.41 - 5.68i)T
23 1−4.79T 1 - 4.79T
good2 1−3.08iT−4T2 1 - 3.08iT - 4T^{2}
3 1+3.11T+9T2 1 + 3.11T + 9T^{2}
5 1−8.57T+25T2 1 - 8.57T + 25T^{2}
7 1−4.55iT−49T2 1 - 4.55iT - 49T^{2}
13 1−5.40iT−169T2 1 - 5.40iT - 169T^{2}
17 1−30.3iT−289T2 1 - 30.3iT - 289T^{2}
19 1+32.2iT−361T2 1 + 32.2iT - 361T^{2}
29 1+3.47iT−841T2 1 + 3.47iT - 841T^{2}
31 1+51.2T+961T2 1 + 51.2T + 961T^{2}
37 1+47.7T+1.36e3T2 1 + 47.7T + 1.36e3T^{2}
41 1−68.9iT−1.68e3T2 1 - 68.9iT - 1.68e3T^{2}
43 1+3.80iT−1.84e3T2 1 + 3.80iT - 1.84e3T^{2}
47 1+55.3T+2.20e3T2 1 + 55.3T + 2.20e3T^{2}
53 1−45.7T+2.80e3T2 1 - 45.7T + 2.80e3T^{2}
59 1−43.9T+3.48e3T2 1 - 43.9T + 3.48e3T^{2}
61 1+17.4iT−3.72e3T2 1 + 17.4iT - 3.72e3T^{2}
67 1−78.2T+4.48e3T2 1 - 78.2T + 4.48e3T^{2}
71 1−15.7T+5.04e3T2 1 - 15.7T + 5.04e3T^{2}
73 1−66.2iT−5.32e3T2 1 - 66.2iT - 5.32e3T^{2}
79 1−66.9iT−6.24e3T2 1 - 66.9iT - 6.24e3T^{2}
83 1+38.9iT−6.88e3T2 1 + 38.9iT - 6.88e3T^{2}
89 1−70.6T+7.92e3T2 1 - 70.6T + 7.92e3T^{2}
97 1−18.9T+9.40e3T2 1 - 18.9T + 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

−12.78942184027934712763236185441, −11.28700154645245989400242354966, −10.35484263417313479667147408652, −9.252706948469484873204621636431, −8.469084332820306287234311479429, −6.92590664920782996150993463584, −6.26339723422382665522198279345, −5.46676864245316316617613341406, −4.97905292984170535141348905873, −2.17960155854401817524238328853, 0.57082405665431678478609623005, 2.00768603141348098634055050725, 3.30777956580342262207839596300, 5.13970465782908961028460223738, 5.68060529945723533621318722615, 7.05848444205688183104029123733, 8.875060055284271929969341219954, 9.931270395932595027697752513269, 10.44194577106220172006192388899, 11.04930134974160605068578260490

Graph of the ZZ-function along the critical line