Properties

Label 2-475-1.1-c1-0-13
Degree $2$
Conductor $475$
Sign $-1$
Analytic cond. $3.79289$
Root an. cond. $1.94753$
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  − 0.311·2-s − 2.90·3-s − 1.90·4-s + 0.903·6-s + 4.42·7-s + 1.21·8-s + 5.42·9-s − 2.62·11-s + 5.52·12-s − 0.474·13-s − 1.37·14-s + 3.42·16-s − 5.05·17-s − 1.68·18-s − 19-s − 12.8·21-s + 0.815·22-s + 1.37·23-s − 3.52·24-s + 0.147·26-s − 7.05·27-s − 8.42·28-s − 7.80·29-s + 1.24·31-s − 3.49·32-s + 7.61·33-s + 1.57·34-s + ⋯
L(s)  = 1  − 0.219·2-s − 1.67·3-s − 0.951·4-s + 0.368·6-s + 1.67·7-s + 0.429·8-s + 1.80·9-s − 0.790·11-s + 1.59·12-s − 0.131·13-s − 0.368·14-s + 0.857·16-s − 1.22·17-s − 0.398·18-s − 0.229·19-s − 2.80·21-s + 0.173·22-s + 0.287·23-s − 0.719·24-s + 0.0289·26-s − 1.35·27-s − 1.59·28-s − 1.44·29-s + 0.223·31-s − 0.617·32-s + 1.32·33-s + 0.269·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 \)
19 \( 1 + T \)
good2 \( 1 + 0.311T + 2T^{2} \)
3 \( 1 + 2.90T + 3T^{2} \)
7 \( 1 - 4.42T + 7T^{2} \)
11 \( 1 + 2.62T + 11T^{2} \)
13 \( 1 + 0.474T + 13T^{2} \)
17 \( 1 + 5.05T + 17T^{2} \)
23 \( 1 - 1.37T + 23T^{2} \)
29 \( 1 + 7.80T + 29T^{2} \)
31 \( 1 - 1.24T + 31T^{2} \)
37 \( 1 + 4.47T + 37T^{2} \)
41 \( 1 + 5.05T + 41T^{2} \)
43 \( 1 + 12.0T + 43T^{2} \)
47 \( 1 - 4.42T + 47T^{2} \)
53 \( 1 + 7.52T + 53T^{2} \)
59 \( 1 + 2.19T + 59T^{2} \)
61 \( 1 - 3.67T + 61T^{2} \)
67 \( 1 - 1.65T + 67T^{2} \)
71 \( 1 - 7.61T + 71T^{2} \)
73 \( 1 - 3.80T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 - 10.6T + 83T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 + 17.8T + 97T^{2} \)
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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

−10.84454561368487534175163428039, −9.910143149772269824359035553464, −8.693550340092958874569697763055, −7.87462289202091224606469301332, −6.81730870589100847445965028483, −5.41800378276575073717815413676, −5.03625291677403996088410488284, −4.21324210655885383639464087734, −1.64756415375780415218626594701, 0, 1.64756415375780415218626594701, 4.21324210655885383639464087734, 5.03625291677403996088410488284, 5.41800378276575073717815413676, 6.81730870589100847445965028483, 7.87462289202091224606469301332, 8.693550340092958874569697763055, 9.910143149772269824359035553464, 10.84454561368487534175163428039

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