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Properties

Label 2-294-1.1-c9-0-44
Degree 22
Conductor 294294
Sign −1-1
Analytic cond. 151.420151.420
Root an. cond. 12.305312.3053
Motivic weight 99
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·2-s + 81·3-s + 256·4-s + 76·5-s − 1.29e3·6-s − 4.09e3·8-s + 6.56e3·9-s − 1.21e3·10-s + 3.83e4·11-s + 2.07e4·12-s − 9.82e4·13-s + 6.15e3·15-s + 6.55e4·16-s − 1.04e5·17-s − 1.04e5·18-s + 4.20e5·19-s + 1.94e4·20-s − 6.14e5·22-s − 1.39e5·23-s − 3.31e5·24-s − 1.94e6·25-s + 1.57e6·26-s + 5.31e5·27-s − 1.91e6·29-s − 9.84e4·30-s + 6.37e6·31-s − 1.04e6·32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.0543·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.0384·10-s + 0.790·11-s + 0.288·12-s − 0.954·13-s + 0.0313·15-s + 1/4·16-s − 0.303·17-s − 0.235·18-s + 0.740·19-s + 0.0271·20-s − 0.558·22-s − 0.103·23-s − 0.204·24-s − 0.997·25-s + 0.674·26-s + 0.192·27-s − 0.503·29-s − 0.0222·30-s + 1.24·31-s − 0.176·32-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=(− Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+9/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: −1-1
Analytic conductor: 151.420151.420
Root analytic conductor: 12.305312.3053
Motivic weight: 99
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 294, ( :9/2), −1)(2,\ 294,\ (\ :9/2),\ -1)

Particular Values

L(5)L(5) == 00
L(12)L(\frac12) == 00
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)
bad2 1+p4T 1 + p^{4} T
3 1−p4T 1 - p^{4} T
7 1 1
good5 1−76T+p9T2 1 - 76 T + p^{9} T^{2}
11 1−38386T+p9T2 1 - 38386 T + p^{9} T^{2}
13 1+98298T+p9T2 1 + 98298 T + p^{9} T^{2}
17 1+104524T+p9T2 1 + 104524 T + p^{9} T^{2}
19 1−420580T+p9T2 1 - 420580 T + p^{9} T^{2}
23 1+139118T+p9T2 1 + 139118 T + p^{9} T^{2}
29 1+1916290T+p9T2 1 + 1916290 T + p^{9} T^{2}
31 1−6379488T+p9T2 1 - 6379488 T + p^{9} T^{2}
37 1+6629278T+p9T2 1 + 6629278 T + p^{9} T^{2}
41 1−6692112T+p9T2 1 - 6692112 T + p^{9} T^{2}
43 1+23269732T+p9T2 1 + 23269732 T + p^{9} T^{2}
47 1−22000596T+p9T2 1 - 22000596 T + p^{9} T^{2}
53 1−18919770T+p9T2 1 - 18919770 T + p^{9} T^{2}
59 1+179035544T+p9T2 1 + 179035544 T + p^{9} T^{2}
61 1−19797786T+p9T2 1 - 19797786 T + p^{9} T^{2}
67 1+263015240T+p9T2 1 + 263015240 T + p^{9} T^{2}
71 1−22447678T+p9T2 1 - 22447678 T + p^{9} T^{2}
73 1+11023774T+p9T2 1 + 11023774 T + p^{9} T^{2}
79 1+284917908T+p9T2 1 + 284917908 T + p^{9} T^{2}
83 1−226865924T+p9T2 1 - 226865924 T + p^{9} T^{2}
89 1−191377296T+p9T2 1 - 191377296 T + p^{9} T^{2}
97 1−1162236578T+p9T2 1 - 1162236578 T + p^{9} T^{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

−9.605083953352980686852875469022, −8.914307098434668118560959751556, −7.85994526955503223168915842830, −7.11604565596641529339698579706, −6.03210364474251277098783725357, −4.63910782466909243693851030773, −3.40538091158149579778656389712, −2.29867280478219999489315965007, −1.29704622966914576993545882125, 0, 1.29704622966914576993545882125, 2.29867280478219999489315965007, 3.40538091158149579778656389712, 4.63910782466909243693851030773, 6.03210364474251277098783725357, 7.11604565596641529339698579706, 7.85994526955503223168915842830, 8.914307098434668118560959751556, 9.605083953352980686852875469022

Graph of the ZZ-function along the critical line