Properties

Label 2-2013-1.1-c3-0-223
Degree $2$
Conductor $2013$
Sign $1$
Analytic cond. $118.770$
Root an. cond. $10.8982$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.23·2-s + 3·3-s + 2.46·4-s + 19.2·5-s + 9.70·6-s + 32.7·7-s − 17.9·8-s + 9·9-s + 62.4·10-s − 11·11-s + 7.40·12-s + 24.7·13-s + 105.·14-s + 57.8·15-s − 77.6·16-s + 60.7·17-s + 29.1·18-s − 22.2·19-s + 47.6·20-s + 98.1·21-s − 35.5·22-s + 113.·23-s − 53.7·24-s + 247.·25-s + 80.0·26-s + 27·27-s + 80.7·28-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.577·3-s + 0.308·4-s + 1.72·5-s + 0.660·6-s + 1.76·7-s − 0.791·8-s + 0.333·9-s + 1.97·10-s − 0.301·11-s + 0.178·12-s + 0.528·13-s + 2.02·14-s + 0.996·15-s − 1.21·16-s + 0.866·17-s + 0.381·18-s − 0.268·19-s + 0.532·20-s + 1.02·21-s − 0.344·22-s + 1.03·23-s − 0.456·24-s + 1.97·25-s + 0.604·26-s + 0.192·27-s + 0.545·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $1$
Analytic conductor: \(118.770\)
Root analytic conductor: \(10.8982\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2013,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(8.704775337\)
\(L(\frac12)\) \(\approx\) \(8.704775337\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
11 \( 1 + 11T \)
61 \( 1 - 61T \)
good2 \( 1 - 3.23T + 8T^{2} \)
5 \( 1 - 19.2T + 125T^{2} \)
7 \( 1 - 32.7T + 343T^{2} \)
13 \( 1 - 24.7T + 2.19e3T^{2} \)
17 \( 1 - 60.7T + 4.91e3T^{2} \)
19 \( 1 + 22.2T + 6.85e3T^{2} \)
23 \( 1 - 113.T + 1.21e4T^{2} \)
29 \( 1 + 294.T + 2.43e4T^{2} \)
31 \( 1 - 117.T + 2.97e4T^{2} \)
37 \( 1 - 261.T + 5.06e4T^{2} \)
41 \( 1 + 287.T + 6.89e4T^{2} \)
43 \( 1 + 551.T + 7.95e4T^{2} \)
47 \( 1 + 165.T + 1.03e5T^{2} \)
53 \( 1 - 89.2T + 1.48e5T^{2} \)
59 \( 1 - 769.T + 2.05e5T^{2} \)
67 \( 1 + 39.7T + 3.00e5T^{2} \)
71 \( 1 - 348.T + 3.57e5T^{2} \)
73 \( 1 + 147.T + 3.89e5T^{2} \)
79 \( 1 + 911.T + 4.93e5T^{2} \)
83 \( 1 - 227.T + 5.71e5T^{2} \)
89 \( 1 + 742.T + 7.04e5T^{2} \)
97 \( 1 + 18.5T + 9.12e5T^{2} \)
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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

−8.726817104025321369256880115807, −8.175500846253522006015611315039, −7.05778034242224502813454281038, −6.10529821084208857172193688953, −5.24229631580895328183691495295, −5.07546917563757594279986139989, −3.94405934685023980666428578891, −2.89050454735013302927065993078, −2.00752754304632570134895983835, −1.27738572957680418007304332358, 1.27738572957680418007304332358, 2.00752754304632570134895983835, 2.89050454735013302927065993078, 3.94405934685023980666428578891, 5.07546917563757594279986139989, 5.24229631580895328183691495295, 6.10529821084208857172193688953, 7.05778034242224502813454281038, 8.175500846253522006015611315039, 8.726817104025321369256880115807

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