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Properties

Label 2-245-1.1-c9-0-28
Degree 22
Conductor 245245
Sign 11
Analytic cond. 126.183126.183
Root an. cond. 11.233111.2331
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 21.3·2-s + 190.·3-s − 57.2·4-s + 625·5-s − 4.05e3·6-s + 1.21e4·8-s + 1.64e4·9-s − 1.33e4·10-s − 7.98e4·11-s − 1.08e4·12-s − 1.48e5·13-s + 1.18e5·15-s − 2.29e5·16-s − 1.96e5·17-s − 3.50e5·18-s − 7.43e5·19-s − 3.58e4·20-s + 1.70e6·22-s + 1.25e6·23-s + 2.30e6·24-s + 3.90e5·25-s + 3.16e6·26-s − 6.16e5·27-s + 4.19e6·29-s − 2.53e6·30-s − 6.78e6·31-s − 1.32e6·32-s + ⋯
L(s)  = 1  − 0.942·2-s + 1.35·3-s − 0.111·4-s + 0.447·5-s − 1.27·6-s + 1.04·8-s + 0.835·9-s − 0.421·10-s − 1.64·11-s − 0.151·12-s − 1.43·13-s + 0.605·15-s − 0.875·16-s − 0.571·17-s − 0.787·18-s − 1.30·19-s − 0.0500·20-s + 1.55·22-s + 0.937·23-s + 1.41·24-s + 0.200·25-s + 1.35·26-s − 0.223·27-s + 1.10·29-s − 0.570·30-s − 1.32·31-s − 0.222·32-s + ⋯

Functional equation

Λ(s)=(245s/2 ΓC(s) L(s)=( Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
Λ(s)=(245s/2 ΓC(s+9/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 245245    =    5⋅725 \cdot 7^{2}
Sign: 11
Analytic conductor: 126.183126.183
Root analytic conductor: 11.233111.2331
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 245, ( :9/2), 1)(2,\ 245,\ (\ :9/2),\ 1)

Particular Values

L(5)L(5) ≈\approx 1.3357094531.335709453
L(12)L(\frac12) ≈\approx 1.3357094531.335709453
L(112)L(\frac{11}{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)
bad5 1−625T 1 - 625T
7 1 1
good2 1+21.3T+512T2 1 + 21.3T + 512T^{2}
3 1−190.T+1.96e4T2 1 - 190.T + 1.96e4T^{2}
11 1+7.98e4T+2.35e9T2 1 + 7.98e4T + 2.35e9T^{2}
13 1+1.48e5T+1.06e10T2 1 + 1.48e5T + 1.06e10T^{2}
17 1+1.96e5T+1.18e11T2 1 + 1.96e5T + 1.18e11T^{2}
19 1+7.43e5T+3.22e11T2 1 + 7.43e5T + 3.22e11T^{2}
23 1−1.25e6T+1.80e12T2 1 - 1.25e6T + 1.80e12T^{2}
29 1−4.19e6T+1.45e13T2 1 - 4.19e6T + 1.45e13T^{2}
31 1+6.78e6T+2.64e13T2 1 + 6.78e6T + 2.64e13T^{2}
37 1−9.11e6T+1.29e14T2 1 - 9.11e6T + 1.29e14T^{2}
41 1−1.68e7T+3.27e14T2 1 - 1.68e7T + 3.27e14T^{2}
43 1−1.19e7T+5.02e14T2 1 - 1.19e7T + 5.02e14T^{2}
47 1−5.48e7T+1.11e15T2 1 - 5.48e7T + 1.11e15T^{2}
53 1−3.28e7T+3.29e15T2 1 - 3.28e7T + 3.29e15T^{2}
59 1−7.27e7T+8.66e15T2 1 - 7.27e7T + 8.66e15T^{2}
61 1−1.26e8T+1.16e16T2 1 - 1.26e8T + 1.16e16T^{2}
67 1−8.96e7T+2.72e16T2 1 - 8.96e7T + 2.72e16T^{2}
71 1−1.94e8T+4.58e16T2 1 - 1.94e8T + 4.58e16T^{2}
73 1+2.22e8T+5.88e16T2 1 + 2.22e8T + 5.88e16T^{2}
79 1−2.27e8T+1.19e17T2 1 - 2.27e8T + 1.19e17T^{2}
83 1+1.48e7T+1.86e17T2 1 + 1.48e7T + 1.86e17T^{2}
89 1+4.47e8T+3.50e17T2 1 + 4.47e8T + 3.50e17T^{2}
97 1−6.78e8T+7.60e17T2 1 - 6.78e8T + 7.60e17T^{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

−10.18306327133243174763228969474, −9.327627759326594224169293412709, −8.657062707340141613661500795380, −7.84109814113098251682989450912, −7.11602023002424458840074167344, −5.29388909936135492761886356124, −4.25490167163713234800179354341, −2.56967521334732639926030776139, −2.22049919164464199768366608518, −0.54634292115906621555041427605, 0.54634292115906621555041427605, 2.22049919164464199768366608518, 2.56967521334732639926030776139, 4.25490167163713234800179354341, 5.29388909936135492761886356124, 7.11602023002424458840074167344, 7.84109814113098251682989450912, 8.657062707340141613661500795380, 9.327627759326594224169293412709, 10.18306327133243174763228969474

Graph of the ZZ-function along the critical line