Properties

Label 2-538-1.1-c1-0-16
Degree $2$
Conductor $538$
Sign $-1$
Analytic cond. $4.29595$
Root an. cond. $2.07266$
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  − 2-s − 0.370·3-s + 4-s + 2.98·5-s + 0.370·6-s − 2.56·7-s − 8-s − 2.86·9-s − 2.98·10-s − 5.77·11-s − 0.370·12-s − 0.953·13-s + 2.56·14-s − 1.10·15-s + 16-s + 2.51·17-s + 2.86·18-s − 7.19·19-s + 2.98·20-s + 0.950·21-s + 5.77·22-s − 5.17·23-s + 0.370·24-s + 3.88·25-s + 0.953·26-s + 2.17·27-s − 2.56·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.213·3-s + 0.5·4-s + 1.33·5-s + 0.151·6-s − 0.969·7-s − 0.353·8-s − 0.954·9-s − 0.942·10-s − 1.74·11-s − 0.106·12-s − 0.264·13-s + 0.685·14-s − 0.285·15-s + 0.250·16-s + 0.610·17-s + 0.674·18-s − 1.65·19-s + 0.666·20-s + 0.207·21-s + 1.23·22-s − 1.07·23-s + 0.0756·24-s + 0.776·25-s + 0.187·26-s + 0.418·27-s − 0.484·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(538\)    =    \(2 \cdot 269\)
Sign: $-1$
Analytic conductor: \(4.29595\)
Root analytic conductor: \(2.07266\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 538,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
269 \( 1 + T \)
good3 \( 1 + 0.370T + 3T^{2} \)
5 \( 1 - 2.98T + 5T^{2} \)
7 \( 1 + 2.56T + 7T^{2} \)
11 \( 1 + 5.77T + 11T^{2} \)
13 \( 1 + 0.953T + 13T^{2} \)
17 \( 1 - 2.51T + 17T^{2} \)
19 \( 1 + 7.19T + 19T^{2} \)
23 \( 1 + 5.17T + 23T^{2} \)
29 \( 1 - 4.63T + 29T^{2} \)
31 \( 1 - 8.16T + 31T^{2} \)
37 \( 1 + 9.43T + 37T^{2} \)
41 \( 1 - 2.46T + 41T^{2} \)
43 \( 1 - 5.86T + 43T^{2} \)
47 \( 1 + 9.50T + 47T^{2} \)
53 \( 1 + 7.51T + 53T^{2} \)
59 \( 1 + 1.01T + 59T^{2} \)
61 \( 1 + 5.86T + 61T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 - 0.650T + 73T^{2} \)
79 \( 1 - 15.5T + 79T^{2} \)
83 \( 1 + 8.27T + 83T^{2} \)
89 \( 1 - 2.89T + 89T^{2} \)
97 \( 1 - 0.956T + 97T^{2} \)
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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

−10.22053309896049561464534792498, −9.753775148186898561456895727304, −8.632210486266637574638671021135, −7.918387694291356414119042394750, −6.45881248104667637849398023658, −6.02725912501540600442617313467, −5.02817614322771125511308549365, −2.99079563057587564427639663242, −2.20527310807920458245245374769, 0, 2.20527310807920458245245374769, 2.99079563057587564427639663242, 5.02817614322771125511308549365, 6.02725912501540600442617313467, 6.45881248104667637849398023658, 7.918387694291356414119042394750, 8.632210486266637574638671021135, 9.753775148186898561456895727304, 10.22053309896049561464534792498

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