Properties

Label 2-354-1.1-c7-0-61
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 + 266.·5-s − 216·6-s + 663.·7-s − 512·8-s + 729·9-s − 2.13e3·10-s − 6.54e3·11-s + 1.72e3·12-s + 5.16e3·13-s − 5.30e3·14-s + 7.20e3·15-s + 4.09e3·16-s + 1.57e4·17-s − 5.83e3·18-s − 4.08e4·19-s + 1.70e4·20-s + 1.79e4·21-s + 5.23e4·22-s + 2.12e4·23-s − 1.38e4·24-s − 6.87e3·25-s − 4.13e4·26-s + 1.96e4·27-s + 4.24e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.954·5-s − 0.408·6-s + 0.730·7-s − 0.353·8-s + 0.333·9-s − 0.675·10-s − 1.48·11-s + 0.288·12-s + 0.651·13-s − 0.516·14-s + 0.551·15-s + 0.250·16-s + 0.778·17-s − 0.235·18-s − 1.36·19-s + 0.477·20-s + 0.421·21-s + 1.04·22-s + 0.363·23-s − 0.204·24-s − 0.0880·25-s − 0.460·26-s + 0.192·27-s + 0.365·28-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 \( 1 - 266.T + 7.81e4T^{2} \)
7 \( 1 - 663.T + 8.23e5T^{2} \)
11 \( 1 + 6.54e3T + 1.94e7T^{2} \)
13 \( 1 - 5.16e3T + 6.27e7T^{2} \)
17 \( 1 - 1.57e4T + 4.10e8T^{2} \)
19 \( 1 + 4.08e4T + 8.93e8T^{2} \)
23 \( 1 - 2.12e4T + 3.40e9T^{2} \)
29 \( 1 + 2.02e5T + 1.72e10T^{2} \)
31 \( 1 + 2.05e5T + 2.75e10T^{2} \)
37 \( 1 + 4.03e5T + 9.49e10T^{2} \)
41 \( 1 + 6.80e4T + 1.94e11T^{2} \)
43 \( 1 - 1.91e4T + 2.71e11T^{2} \)
47 \( 1 + 7.39e5T + 5.06e11T^{2} \)
53 \( 1 - 1.30e5T + 1.17e12T^{2} \)
61 \( 1 - 4.20e5T + 3.14e12T^{2} \)
67 \( 1 + 2.66e6T + 6.06e12T^{2} \)
71 \( 1 - 5.31e6T + 9.09e12T^{2} \)
73 \( 1 - 3.42e6T + 1.10e13T^{2} \)
79 \( 1 + 7.69e6T + 1.92e13T^{2} \)
83 \( 1 + 4.62e6T + 2.71e13T^{2} \)
89 \( 1 + 1.70e6T + 4.42e13T^{2} \)
97 \( 1 - 1.15e7T + 8.07e13T^{2} \)
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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

−9.800159053833421619935540704038, −8.813527586950388544629112789985, −8.103362398172525674187950174245, −7.25930173664682717769862462574, −5.94537528603620129880126214345, −5.09470895382021627569129184424, −3.48494799077439790119742572845, −2.19041468972857325583916451421, −1.60605813361317420938808200797, 0, 1.60605813361317420938808200797, 2.19041468972857325583916451421, 3.48494799077439790119742572845, 5.09470895382021627569129184424, 5.94537528603620129880126214345, 7.25930173664682717769862462574, 8.103362398172525674187950174245, 8.813527586950388544629112789985, 9.800159053833421619935540704038

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