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Properties

Label 2-253-11.10-c2-0-16
Degree 22
Conductor 253253
Sign −0.699−0.714i-0.699 - 0.714i
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  + 2.50i·2-s + 5.31·3-s − 2.28·4-s − 8.14·5-s + 13.3i·6-s + 2.32i·7-s + 4.30i·8-s + 19.2·9-s − 20.4i·10-s + (7.69 + 7.86i)11-s − 12.1·12-s + 19.3i·13-s − 5.81·14-s − 43.2·15-s − 19.9·16-s − 26.1i·17-s + ⋯
L(s)  = 1  + 1.25i·2-s + 1.77·3-s − 0.570·4-s − 1.62·5-s + 2.22i·6-s + 0.331i·7-s + 0.538i·8-s + 2.13·9-s − 2.04i·10-s + (0.699 + 0.714i)11-s − 1.01·12-s + 1.48i·13-s − 0.415·14-s − 2.88·15-s − 1.24·16-s − 1.53i·17-s + ⋯

Functional equation

Λ(s)=(253s/2 ΓC(s) L(s)=((−0.699−0.714i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(253s/2 ΓC(s+1) L(s)=((−0.699−0.714i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 253253    =    11⋅2311 \cdot 23
Sign: −0.699−0.714i-0.699 - 0.714i
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), −0.699−0.714i)(2,\ 253,\ (\ :1),\ -0.699 - 0.714i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.890319+2.11688i0.890319 + 2.11688i
L(12)L(\frac12) ≈\approx 0.890319+2.11688i0.890319 + 2.11688i
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+(−7.69−7.86i)T 1 + (-7.69 - 7.86i)T
23 1+4.79T 1 + 4.79T
good2 1−2.50iT−4T2 1 - 2.50iT - 4T^{2}
3 1−5.31T+9T2 1 - 5.31T + 9T^{2}
5 1+8.14T+25T2 1 + 8.14T + 25T^{2}
7 1−2.32iT−49T2 1 - 2.32iT - 49T^{2}
13 1−19.3iT−169T2 1 - 19.3iT - 169T^{2}
17 1+26.1iT−289T2 1 + 26.1iT - 289T^{2}
19 1−7.39iT−361T2 1 - 7.39iT - 361T^{2}
29 1+21.0iT−841T2 1 + 21.0iT - 841T^{2}
31 1−30.2T+961T2 1 - 30.2T + 961T^{2}
37 1−13.8T+1.36e3T2 1 - 13.8T + 1.36e3T^{2}
41 1+30.0iT−1.68e3T2 1 + 30.0iT - 1.68e3T^{2}
43 1+46.1iT−1.84e3T2 1 + 46.1iT - 1.84e3T^{2}
47 1+48.3T+2.20e3T2 1 + 48.3T + 2.20e3T^{2}
53 1+1.13T+2.80e3T2 1 + 1.13T + 2.80e3T^{2}
59 1−39.8T+3.48e3T2 1 - 39.8T + 3.48e3T^{2}
61 1−1.89iT−3.72e3T2 1 - 1.89iT - 3.72e3T^{2}
67 1−102.T+4.48e3T2 1 - 102.T + 4.48e3T^{2}
71 1−49.8T+5.04e3T2 1 - 49.8T + 5.04e3T^{2}
73 1−59.8iT−5.32e3T2 1 - 59.8iT - 5.32e3T^{2}
79 1+88.4iT−6.24e3T2 1 + 88.4iT - 6.24e3T^{2}
83 1+135.iT−6.88e3T2 1 + 135. iT - 6.88e3T^{2}
89 1+144.T+7.92e3T2 1 + 144.T + 7.92e3T^{2}
97 1+70.9T+9.40e3T2 1 + 70.9T + 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

−12.09134497795918491328802362858, −11.55530330369234034205248064327, −9.627282933948991772427892021429, −8.850058972109919690911569662437, −8.141510359458285501444807780191, −7.30415432636394492365815984329, −6.81965339818712239436725628499, −4.64532142321055983614402079052, −3.85770645624123102001027646231, −2.34341021226109657638981022335, 1.08764924655233489092839134305, 2.87722652645863949132567241150, 3.57094998948290198793304565761, 4.21257678042299103521895625796, 6.83637052941288898328376102951, 8.125857272042977522986899235993, 8.316805899768466941189248949048, 9.623886981881320243453453593398, 10.59741427894985310870383242108, 11.37415723534373899109338090153

Graph of the ZZ-function along the critical line