Properties

Label 2-299-1.1-c1-0-4
Degree $2$
Conductor $299$
Sign $1$
Analytic cond. $2.38752$
Root an. cond. $1.54516$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.79·2-s − 2.79·3-s + 5.79·4-s + 3.79·5-s + 7.79·6-s + 7-s − 10.5·8-s + 4.79·9-s − 10.5·10-s − 0.791·11-s − 16.1·12-s + 13-s − 2.79·14-s − 10.5·15-s + 17.9·16-s − 0.791·17-s − 13.3·18-s + 0.582·19-s + 21.9·20-s − 2.79·21-s + 2.20·22-s − 23-s + 29.5·24-s + 9.37·25-s − 2.79·26-s − 4.99·27-s + 5.79·28-s + ⋯
L(s)  = 1  − 1.97·2-s − 1.61·3-s + 2.89·4-s + 1.69·5-s + 3.18·6-s + 0.377·7-s − 3.74·8-s + 1.59·9-s − 3.34·10-s − 0.238·11-s − 4.66·12-s + 0.277·13-s − 0.746·14-s − 2.73·15-s + 4.48·16-s − 0.191·17-s − 3.15·18-s + 0.133·19-s + 4.90·20-s − 0.609·21-s + 0.470·22-s − 0.208·23-s + 6.02·24-s + 1.87·25-s − 0.547·26-s − 0.962·27-s + 1.09·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(299\)    =    \(13 \cdot 23\)
Sign: $1$
Analytic conductor: \(2.38752\)
Root analytic conductor: \(1.54516\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 299,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4569990983\)
\(L(\frac12)\) \(\approx\) \(0.4569990983\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 - T \)
23 \( 1 + T \)
good2 \( 1 + 2.79T + 2T^{2} \)
3 \( 1 + 2.79T + 3T^{2} \)
5 \( 1 - 3.79T + 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 0.791T + 11T^{2} \)
17 \( 1 + 0.791T + 17T^{2} \)
19 \( 1 - 0.582T + 19T^{2} \)
29 \( 1 + 3.20T + 29T^{2} \)
31 \( 1 - 3.20T + 31T^{2} \)
37 \( 1 - 11.5T + 37T^{2} \)
41 \( 1 - 0.582T + 41T^{2} \)
43 \( 1 + 2.37T + 43T^{2} \)
47 \( 1 - 3.79T + 47T^{2} \)
53 \( 1 - 7T + 53T^{2} \)
59 \( 1 + 4.16T + 59T^{2} \)
61 \( 1 - 5T + 61T^{2} \)
67 \( 1 - 5T + 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 - 8.20T + 73T^{2} \)
79 \( 1 + 3.79T + 79T^{2} \)
83 \( 1 - 12.9T + 83T^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 + 4.58T + 97T^{2} \)
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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

−11.22113375094048685682718852660, −10.68224783128585793691876666740, −9.902096644067853016212323238474, −9.289028035625784724535726691343, −8.010030022447235861942561285473, −6.74816789051442958766134962695, −6.13606497772836372235964294502, −5.35043650678296737410816224885, −2.26581472449932792214633875264, −1.04716503990335394306377205145, 1.04716503990335394306377205145, 2.26581472449932792214633875264, 5.35043650678296737410816224885, 6.13606497772836372235964294502, 6.74816789051442958766134962695, 8.010030022447235861942561285473, 9.289028035625784724535726691343, 9.902096644067853016212323238474, 10.68224783128585793691876666740, 11.22113375094048685682718852660

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