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Properties

Label 2-253-11.10-c2-0-41
Degree 22
Conductor 253253
Sign −0.910+0.413i-0.910 + 0.413i
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  − 1.15i·2-s + 2.03·3-s + 2.66·4-s − 8.63·5-s − 2.34i·6-s − 11.4i·7-s − 7.70i·8-s − 4.87·9-s + 9.97i·10-s + (−10.0 + 4.54i)11-s + 5.41·12-s + 13.8i·13-s − 13.2·14-s − 17.5·15-s + 1.75·16-s − 15.2i·17-s + ⋯
L(s)  = 1  − 0.577i·2-s + 0.677·3-s + 0.666·4-s − 1.72·5-s − 0.391i·6-s − 1.64i·7-s − 0.962i·8-s − 0.541·9-s + 0.997i·10-s + (−0.910 + 0.413i)11-s + 0.451·12-s + 1.06i·13-s − 0.948·14-s − 1.16·15-s + 0.109·16-s − 0.896i·17-s + ⋯

Functional equation

Λ(s)=(253s/2 ΓC(s) L(s)=((−0.910+0.413i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 + 0.413i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(253s/2 ΓC(s+1) L(s)=((−0.910+0.413i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 253253    =    11⋅2311 \cdot 23
Sign: −0.910+0.413i-0.910 + 0.413i
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), −0.910+0.413i)(2,\ 253,\ (\ :1),\ -0.910 + 0.413i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.248156−1.14679i0.248156 - 1.14679i
L(12)L(\frac12) ≈\approx 0.248156−1.14679i0.248156 - 1.14679i
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+(10.0−4.54i)T 1 + (10.0 - 4.54i)T
23 1−4.79T 1 - 4.79T
good2 1+1.15iT−4T2 1 + 1.15iT - 4T^{2}
3 1−2.03T+9T2 1 - 2.03T + 9T^{2}
5 1+8.63T+25T2 1 + 8.63T + 25T^{2}
7 1+11.4iT−49T2 1 + 11.4iT - 49T^{2}
13 1−13.8iT−169T2 1 - 13.8iT - 169T^{2}
17 1+15.2iT−289T2 1 + 15.2iT - 289T^{2}
19 1+20.7iT−361T2 1 + 20.7iT - 361T^{2}
29 1+3.94iT−841T2 1 + 3.94iT - 841T^{2}
31 1−23.2T+961T2 1 - 23.2T + 961T^{2}
37 1−8.15T+1.36e3T2 1 - 8.15T + 1.36e3T^{2}
41 1+57.8iT−1.68e3T2 1 + 57.8iT - 1.68e3T^{2}
43 1−26.7iT−1.84e3T2 1 - 26.7iT - 1.84e3T^{2}
47 1−33.6T+2.20e3T2 1 - 33.6T + 2.20e3T^{2}
53 1+36.1T+2.80e3T2 1 + 36.1T + 2.80e3T^{2}
59 1−75.7T+3.48e3T2 1 - 75.7T + 3.48e3T^{2}
61 1+8.02iT−3.72e3T2 1 + 8.02iT - 3.72e3T^{2}
67 1−87.9T+4.48e3T2 1 - 87.9T + 4.48e3T^{2}
71 1+92.6T+5.04e3T2 1 + 92.6T + 5.04e3T^{2}
73 1−29.3iT−5.32e3T2 1 - 29.3iT - 5.32e3T^{2}
79 1+59.4iT−6.24e3T2 1 + 59.4iT - 6.24e3T^{2}
83 1−138.iT−6.88e3T2 1 - 138. iT - 6.88e3T^{2}
89 1−103.T+7.92e3T2 1 - 103.T + 7.92e3T^{2}
97 1+0.294T+9.40e3T2 1 + 0.294T + 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

−11.28996867588087863485119028505, −10.81858332346539464171007040902, −9.611114972535137210753534614827, −8.285719584314956869158737690626, −7.32452907911919048961393137115, −7.00789836602816516380089459930, −4.57532517162768516512494890044, −3.69963751770568574543801711943, −2.65788896685577646673971452066, −0.53007661579108166128813225438, 2.58129185142838703156570745255, 3.38918379580458723295999237943, 5.28597590580298054013400735640, 6.14025135347473913600510209914, 7.78935859365538656998239851622, 8.120543445770525780130191525937, 8.703843800992040330185163005712, 10.49481150467019142227437390282, 11.47903898955260195994098009310, 12.08111097593651240780614013565

Graph of the ZZ-function along the critical line