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Properties

Label 2-13e2-1.1-c7-0-42
Degree 22
Conductor 169169
Sign −1-1
Analytic cond. 52.793052.7930
Root an. cond. 7.265887.26588
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 21.5·2-s + 28.3·3-s + 338.·4-s − 473.·5-s − 612.·6-s + 99.1·7-s − 4.54e3·8-s − 1.38e3·9-s + 1.02e4·10-s + 3.43e3·11-s + 9.59e3·12-s − 2.14e3·14-s − 1.34e4·15-s + 5.47e4·16-s + 2.28e4·17-s + 2.98e4·18-s + 3.69e3·19-s − 1.60e5·20-s + 2.81e3·21-s − 7.41e4·22-s − 6.89e4·23-s − 1.28e5·24-s + 1.46e5·25-s − 1.01e5·27-s + 3.35e4·28-s − 1.42e4·29-s + 2.90e5·30-s + ⋯
L(s)  = 1  − 1.90·2-s + 0.606·3-s + 2.64·4-s − 1.69·5-s − 1.15·6-s + 0.109·7-s − 3.13·8-s − 0.631·9-s + 3.23·10-s + 0.777·11-s + 1.60·12-s − 0.208·14-s − 1.02·15-s + 3.34·16-s + 1.13·17-s + 1.20·18-s + 0.123·19-s − 4.47·20-s + 0.0663·21-s − 1.48·22-s − 1.18·23-s − 1.90·24-s + 1.87·25-s − 0.990·27-s + 0.288·28-s − 0.108·29-s + 1.96·30-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: −1-1
Analytic conductor: 52.793052.7930
Root analytic conductor: 7.265887.26588
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 169, ( :7/2), −1)(2,\ 169,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
L(92)L(\frac{9}{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)
bad13 1 1
good2 1+21.5T+128T2 1 + 21.5T + 128T^{2}
3 1−28.3T+2.18e3T2 1 - 28.3T + 2.18e3T^{2}
5 1+473.T+7.81e4T2 1 + 473.T + 7.81e4T^{2}
7 1−99.1T+8.23e5T2 1 - 99.1T + 8.23e5T^{2}
11 1−3.43e3T+1.94e7T2 1 - 3.43e3T + 1.94e7T^{2}
17 1−2.28e4T+4.10e8T2 1 - 2.28e4T + 4.10e8T^{2}
19 1−3.69e3T+8.93e8T2 1 - 3.69e3T + 8.93e8T^{2}
23 1+6.89e4T+3.40e9T2 1 + 6.89e4T + 3.40e9T^{2}
29 1+1.42e4T+1.72e10T2 1 + 1.42e4T + 1.72e10T^{2}
31 1−6.63e4T+2.75e10T2 1 - 6.63e4T + 2.75e10T^{2}
37 1−4.17e5T+9.49e10T2 1 - 4.17e5T + 9.49e10T^{2}
41 1−4.64e5T+1.94e11T2 1 - 4.64e5T + 1.94e11T^{2}
43 1−4.13e5T+2.71e11T2 1 - 4.13e5T + 2.71e11T^{2}
47 1−2.87e5T+5.06e11T2 1 - 2.87e5T + 5.06e11T^{2}
53 1−4.48e5T+1.17e12T2 1 - 4.48e5T + 1.17e12T^{2}
59 1+2.73e6T+2.48e12T2 1 + 2.73e6T + 2.48e12T^{2}
61 1−1.80e6T+3.14e12T2 1 - 1.80e6T + 3.14e12T^{2}
67 1+2.14e6T+6.06e12T2 1 + 2.14e6T + 6.06e12T^{2}
71 1+4.21e6T+9.09e12T2 1 + 4.21e6T + 9.09e12T^{2}
73 1−4.79e6T+1.10e13T2 1 - 4.79e6T + 1.10e13T^{2}
79 1−9.39e5T+1.92e13T2 1 - 9.39e5T + 1.92e13T^{2}
83 1+4.55e5T+2.71e13T2 1 + 4.55e5T + 2.71e13T^{2}
89 1−8.34e5T+4.42e13T2 1 - 8.34e5T + 4.42e13T^{2}
97 1−4.92e6T+8.07e13T2 1 - 4.92e6T + 8.07e13T^{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

−10.91063470224244438098741322025, −9.660019875812038829040561426008, −8.792469215521165731948368212463, −7.88439249714487204854987188955, −7.61507757257929170835361911316, −6.17505930660187662406110104419, −3.82040934918995136057647932000, −2.70900190793780142010406581558, −1.09807136588851126545503552799, 0, 1.09807136588851126545503552799, 2.70900190793780142010406581558, 3.82040934918995136057647932000, 6.17505930660187662406110104419, 7.61507757257929170835361911316, 7.88439249714487204854987188955, 8.792469215521165731948368212463, 9.660019875812038829040561426008, 10.91063470224244438098741322025

Graph of the ZZ-function along the critical line