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Properties

Label 2-130-1.1-c1-0-2
Degree 22
Conductor 130130
Sign 11
Analytic cond. 1.038051.03805
Root an. cond. 1.018841.01884
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s + 4-s − 5-s + 2·6-s − 4·7-s + 8-s + 9-s − 10-s − 2·11-s + 2·12-s − 13-s − 4·14-s − 2·15-s + 16-s + 2·17-s + 18-s + 6·19-s − 20-s − 8·21-s − 2·22-s + 6·23-s + 2·24-s + 25-s − 26-s − 4·27-s − 4·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.577·12-s − 0.277·13-s − 1.06·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s − 1.74·21-s − 0.426·22-s + 1.25·23-s + 0.408·24-s + 1/5·25-s − 0.196·26-s − 0.769·27-s − 0.755·28-s + ⋯

Functional equation

Λ(s)=(130s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(130s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 130130    =    2⋅5⋅132 \cdot 5 \cdot 13
Sign: 11
Analytic conductor: 1.038051.03805
Root analytic conductor: 1.018841.01884
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 130, ( :1/2), 1)(2,\ 130,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7730949031.773094903
L(12)L(\frac12) ≈\approx 1.7730949031.773094903
L(32)L(\frac{3}{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)Isogeny Class over Fp\mathbf{F}_p
bad2 1−T 1 - T
5 1+T 1 + T
13 1+T 1 + T
good3 1−2T+pT2 1 - 2 T + p T^{2} 1.3.ac
7 1+4T+pT2 1 + 4 T + p T^{2} 1.7.e
11 1+2T+pT2 1 + 2 T + p T^{2} 1.11.c
17 1−2T+pT2 1 - 2 T + p T^{2} 1.17.ac
19 1−6T+pT2 1 - 6 T + p T^{2} 1.19.ag
23 1−6T+pT2 1 - 6 T + p T^{2} 1.23.ag
29 1−2T+pT2 1 - 2 T + p T^{2} 1.29.ac
31 1+6T+pT2 1 + 6 T + p T^{2} 1.31.g
37 1+2T+pT2 1 + 2 T + p T^{2} 1.37.c
41 1−10T+pT2 1 - 10 T + p T^{2} 1.41.ak
43 1+10T+pT2 1 + 10 T + p T^{2} 1.43.k
47 1+12T+pT2 1 + 12 T + p T^{2} 1.47.m
53 1−2T+pT2 1 - 2 T + p T^{2} 1.53.ac
59 1−10T+pT2 1 - 10 T + p T^{2} 1.59.ak
61 1−2T+pT2 1 - 2 T + p T^{2} 1.61.ac
67 1+12T+pT2 1 + 12 T + p T^{2} 1.67.m
71 1−10T+pT2 1 - 10 T + p T^{2} 1.71.ak
73 1−10T+pT2 1 - 10 T + p T^{2} 1.73.ak
79 1+4T+pT2 1 + 4 T + p T^{2} 1.79.e
83 1+pT2 1 + p T^{2} 1.83.a
89 1+14T+pT2 1 + 14 T + p T^{2} 1.89.o
97 1−14T+pT2 1 - 14 T + p T^{2} 1.97.ao
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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

−13.26679558606257756188931391942, −12.71202338625229541571828648444, −11.47889106396016028805556897933, −10.04470252993954468916012522294, −9.161137327460865013114506016469, −7.83531743824800124245932651769, −6.87551315707654393130092819540, −5.33976892050247260934145252130, −3.53009310754873249134401547682, −2.88253971705917726242541361642, 2.88253971705917726242541361642, 3.53009310754873249134401547682, 5.33976892050247260934145252130, 6.87551315707654393130092819540, 7.83531743824800124245932651769, 9.161137327460865013114506016469, 10.04470252993954468916012522294, 11.47889106396016028805556897933, 12.71202338625229541571828648444, 13.26679558606257756188931391942

Graph of the ZZ-function along the critical line