Properties

Label 2-354-1.1-c7-0-40
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 + 253.·5-s + 216·6-s − 755.·7-s − 512·8-s + 729·9-s − 2.02e3·10-s − 866.·11-s − 1.72e3·12-s + 7.07e3·13-s + 6.04e3·14-s − 6.83e3·15-s + 4.09e3·16-s + 1.32e4·17-s − 5.83e3·18-s − 5.04e4·19-s + 1.62e4·20-s + 2.04e4·21-s + 6.93e3·22-s + 4.07e4·23-s + 1.38e4·24-s − 1.40e4·25-s − 5.66e4·26-s − 1.96e4·27-s − 4.83e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.905·5-s + 0.408·6-s − 0.832·7-s − 0.353·8-s + 0.333·9-s − 0.640·10-s − 0.196·11-s − 0.288·12-s + 0.893·13-s + 0.588·14-s − 0.522·15-s + 0.250·16-s + 0.656·17-s − 0.235·18-s − 1.68·19-s + 0.452·20-s + 0.480·21-s + 0.138·22-s + 0.698·23-s + 0.204·24-s − 0.179·25-s − 0.631·26-s − 0.192·27-s − 0.416·28-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 \( 1 - 253.T + 7.81e4T^{2} \)
7 \( 1 + 755.T + 8.23e5T^{2} \)
11 \( 1 + 866.T + 1.94e7T^{2} \)
13 \( 1 - 7.07e3T + 6.27e7T^{2} \)
17 \( 1 - 1.32e4T + 4.10e8T^{2} \)
19 \( 1 + 5.04e4T + 8.93e8T^{2} \)
23 \( 1 - 4.07e4T + 3.40e9T^{2} \)
29 \( 1 - 2.51e4T + 1.72e10T^{2} \)
31 \( 1 - 1.58e5T + 2.75e10T^{2} \)
37 \( 1 + 3.41e5T + 9.49e10T^{2} \)
41 \( 1 - 5.09e4T + 1.94e11T^{2} \)
43 \( 1 + 1.33e5T + 2.71e11T^{2} \)
47 \( 1 + 2.71e5T + 5.06e11T^{2} \)
53 \( 1 + 7.71e5T + 1.17e12T^{2} \)
61 \( 1 - 1.80e6T + 3.14e12T^{2} \)
67 \( 1 - 2.02e6T + 6.06e12T^{2} \)
71 \( 1 - 1.14e6T + 9.09e12T^{2} \)
73 \( 1 - 6.48e6T + 1.10e13T^{2} \)
79 \( 1 + 3.37e6T + 1.92e13T^{2} \)
83 \( 1 - 3.34e6T + 2.71e13T^{2} \)
89 \( 1 - 9.38e6T + 4.42e13T^{2} \)
97 \( 1 + 1.22e7T + 8.07e13T^{2} \)
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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

−9.945525342955458729050226797002, −9.019165492075934659971980123946, −8.077446903231495049403052683712, −6.61878288172380372307203775043, −6.28448653088672878438615158867, −5.17788825719142921423402525937, −3.61981895236020346874125092312, −2.30665041566940109513870408851, −1.16979432769588098364941850424, 0, 1.16979432769588098364941850424, 2.30665041566940109513870408851, 3.61981895236020346874125092312, 5.17788825719142921423402525937, 6.28448653088672878438615158867, 6.61878288172380372307203775043, 8.077446903231495049403052683712, 9.019165492075934659971980123946, 9.945525342955458729050226797002

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