Properties

Label 2-538-1.1-c1-0-19
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 − 1.61·3-s + 4-s − 1.38·5-s − 1.61·6-s + 0.236·7-s + 8-s − 0.381·9-s − 1.38·10-s − 4.23·11-s − 1.61·12-s − 13-s + 0.236·14-s + 2.23·15-s + 16-s − 4.23·17-s − 0.381·18-s + 2·19-s − 1.38·20-s − 0.381·21-s − 4.23·22-s − 3.76·23-s − 1.61·24-s − 3.09·25-s − 26-s + 5.47·27-s + 0.236·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.934·3-s + 0.5·4-s − 0.618·5-s − 0.660·6-s + 0.0892·7-s + 0.353·8-s − 0.127·9-s − 0.437·10-s − 1.27·11-s − 0.467·12-s − 0.277·13-s + 0.0630·14-s + 0.577·15-s + 0.250·16-s − 1.02·17-s − 0.0900·18-s + 0.458·19-s − 0.309·20-s − 0.0833·21-s − 0.903·22-s − 0.784·23-s − 0.330·24-s − 0.618·25-s − 0.196·26-s + 1.05·27-s + 0.0446·28-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 \( 1 + 1.61T + 3T^{2} \)
5 \( 1 + 1.38T + 5T^{2} \)
7 \( 1 - 0.236T + 7T^{2} \)
11 \( 1 + 4.23T + 11T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + 4.23T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 3.76T + 23T^{2} \)
29 \( 1 + 5.85T + 29T^{2} \)
31 \( 1 + 1.47T + 31T^{2} \)
37 \( 1 - 4.70T + 37T^{2} \)
41 \( 1 + 5.70T + 41T^{2} \)
43 \( 1 + 2.23T + 43T^{2} \)
47 \( 1 + 0.145T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 13.5T + 59T^{2} \)
61 \( 1 - 3.47T + 61T^{2} \)
67 \( 1 - 6.23T + 67T^{2} \)
71 \( 1 + 3.70T + 71T^{2} \)
73 \( 1 - 2.70T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + 0.763T + 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 + 18.6T + 97T^{2} \)
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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

−10.75550022492265920355365148009, −9.779310370633365036992065654641, −8.340542059575305887345664585683, −7.55313253354201825755994795926, −6.52286239153267665937487243535, −5.53522707827406947590541907383, −4.88465287099634814764121847312, −3.72426930725569106217054384489, −2.34630354444356126931490073198, 0, 2.34630354444356126931490073198, 3.72426930725569106217054384489, 4.88465287099634814764121847312, 5.53522707827406947590541907383, 6.52286239153267665937487243535, 7.55313253354201825755994795926, 8.340542059575305887345664585683, 9.779310370633365036992065654641, 10.75550022492265920355365148009

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