Properties

Label 2-354-1.1-c7-0-54
Degree $2$
Conductor $354$
Sign $-1$
Analytic cond. $110.584$
Root an. cond. $10.5159$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 27·3-s + 64·4-s + 176.·5-s − 216·6-s − 148.·7-s − 512·8-s + 729·9-s − 1.41e3·10-s + 994.·11-s + 1.72e3·12-s − 1.09e4·13-s + 1.18e3·14-s + 4.76e3·15-s + 4.09e3·16-s + 9.14e3·17-s − 5.83e3·18-s + 3.32e4·19-s + 1.12e4·20-s − 4.00e3·21-s − 7.95e3·22-s − 2.28e4·23-s − 1.38e4·24-s − 4.70e4·25-s + 8.78e4·26-s + 1.96e4·27-s − 9.49e3·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.631·5-s − 0.408·6-s − 0.163·7-s − 0.353·8-s + 0.333·9-s − 0.446·10-s + 0.225·11-s + 0.288·12-s − 1.38·13-s + 0.115·14-s + 0.364·15-s + 0.250·16-s + 0.451·17-s − 0.235·18-s + 1.11·19-s + 0.315·20-s − 0.0943·21-s − 0.159·22-s − 0.390·23-s − 0.204·24-s − 0.601·25-s + 0.980·26-s + 0.192·27-s − 0.0817·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(110.584\)
Root analytic conductor: \(10.5159\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 354,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 8T \)
3 \( 1 - 27T \)
59 \( 1 + 2.05e5T \)
good5 \( 1 - 176.T + 7.81e4T^{2} \)
7 \( 1 + 148.T + 8.23e5T^{2} \)
11 \( 1 - 994.T + 1.94e7T^{2} \)
13 \( 1 + 1.09e4T + 6.27e7T^{2} \)
17 \( 1 - 9.14e3T + 4.10e8T^{2} \)
19 \( 1 - 3.32e4T + 8.93e8T^{2} \)
23 \( 1 + 2.28e4T + 3.40e9T^{2} \)
29 \( 1 + 2.05e5T + 1.72e10T^{2} \)
31 \( 1 - 2.34e5T + 2.75e10T^{2} \)
37 \( 1 + 1.70e5T + 9.49e10T^{2} \)
41 \( 1 + 7.54e5T + 1.94e11T^{2} \)
43 \( 1 - 4.81e5T + 2.71e11T^{2} \)
47 \( 1 - 4.61e5T + 5.06e11T^{2} \)
53 \( 1 + 8.28e5T + 1.17e12T^{2} \)
61 \( 1 + 4.68e5T + 3.14e12T^{2} \)
67 \( 1 + 4.04e6T + 6.06e12T^{2} \)
71 \( 1 - 1.34e6T + 9.09e12T^{2} \)
73 \( 1 + 2.27e6T + 1.10e13T^{2} \)
79 \( 1 - 4.13e6T + 1.92e13T^{2} \)
83 \( 1 - 2.97e6T + 2.71e13T^{2} \)
89 \( 1 - 2.24e6T + 4.42e13T^{2} \)
97 \( 1 + 5.71e6T + 8.07e13T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.730456349868614047570747017826, −9.113595496355845029914584627069, −7.892446476793421500021777091944, −7.26183275104742251929163101896, −6.09515979355996916705850098847, −4.98049692652732538738593554217, −3.45059688085521716860597355868, −2.38951863956471379123836920100, −1.42977823233667579174145630406, 0, 1.42977823233667579174145630406, 2.38951863956471379123836920100, 3.45059688085521716860597355868, 4.98049692652732538738593554217, 6.09515979355996916705850098847, 7.26183275104742251929163101896, 7.892446476793421500021777091944, 9.113595496355845029914584627069, 9.730456349868614047570747017826

Graph of the $Z$-function along the critical line