Properties

Label 2-476-1.1-c1-0-6
Degree $2$
Conductor $476$
Sign $-1$
Analytic cond. $3.80087$
Root an. cond. $1.94958$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.302·3-s − 3.30·5-s + 7-s − 2.90·9-s + 4.60·11-s − 6.60·13-s − 1.00·15-s − 17-s − 6·19-s + 0.302·21-s − 2.60·23-s + 5.90·25-s − 1.78·27-s − 8.60·29-s − 6.69·31-s + 1.39·33-s − 3.30·35-s + 7.21·37-s − 2.00·39-s + 9.51·41-s − 4.30·43-s + 9.60·45-s + 10·47-s + 49-s − 0.302·51-s + 0.697·53-s − 15.2·55-s + ⋯
L(s)  = 1  + 0.174·3-s − 1.47·5-s + 0.377·7-s − 0.969·9-s + 1.38·11-s − 1.83·13-s − 0.258·15-s − 0.242·17-s − 1.37·19-s + 0.0660·21-s − 0.543·23-s + 1.18·25-s − 0.344·27-s − 1.59·29-s − 1.20·31-s + 0.242·33-s − 0.558·35-s + 1.18·37-s − 0.320·39-s + 1.48·41-s − 0.656·43-s + 1.43·45-s + 1.45·47-s + 0.142·49-s − 0.0423·51-s + 0.0957·53-s − 2.05·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $-1$
Analytic conductor: \(3.80087\)
Root analytic conductor: \(1.94958\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 476,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
17 \( 1 + T \)
good3 \( 1 - 0.302T + 3T^{2} \)
5 \( 1 + 3.30T + 5T^{2} \)
11 \( 1 - 4.60T + 11T^{2} \)
13 \( 1 + 6.60T + 13T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + 2.60T + 23T^{2} \)
29 \( 1 + 8.60T + 29T^{2} \)
31 \( 1 + 6.69T + 31T^{2} \)
37 \( 1 - 7.21T + 37T^{2} \)
41 \( 1 - 9.51T + 41T^{2} \)
43 \( 1 + 4.30T + 43T^{2} \)
47 \( 1 - 10T + 47T^{2} \)
53 \( 1 - 0.697T + 53T^{2} \)
59 \( 1 - 5.21T + 59T^{2} \)
61 \( 1 + 4.30T + 61T^{2} \)
67 \( 1 - 2.69T + 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + 13.5T + 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 - 9.21T + 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + 2.09T + 97T^{2} \)
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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

−10.85696882384837336755774984979, −9.440023554094045298409228756394, −8.739793279493932851068109336666, −7.76481959306128598922806051066, −7.15387354484516515755702646157, −5.85128632622433463351478609513, −4.47554244256704604327094464595, −3.80293690894226375210320066440, −2.32822010793954904243592174695, 0, 2.32822010793954904243592174695, 3.80293690894226375210320066440, 4.47554244256704604327094464595, 5.85128632622433463351478609513, 7.15387354484516515755702646157, 7.76481959306128598922806051066, 8.739793279493932851068109336666, 9.440023554094045298409228756394, 10.85696882384837336755774984979

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