Properties

Label 2-2013-1.1-c3-0-159
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  − 2.38·2-s + 3·3-s − 2.32·4-s + 14.5·5-s − 7.14·6-s + 22.4·7-s + 24.5·8-s + 9·9-s − 34.6·10-s + 11·11-s − 6.96·12-s + 58.9·13-s − 53.4·14-s + 43.5·15-s − 40.0·16-s + 12.2·17-s − 21.4·18-s − 99.5·19-s − 33.7·20-s + 67.3·21-s − 26.2·22-s + 41.8·23-s + 73.7·24-s + 86.0·25-s − 140.·26-s + 27·27-s − 52.0·28-s + ⋯
L(s)  = 1  − 0.842·2-s + 0.577·3-s − 0.290·4-s + 1.29·5-s − 0.486·6-s + 1.21·7-s + 1.08·8-s + 0.333·9-s − 1.09·10-s + 0.301·11-s − 0.167·12-s + 1.25·13-s − 1.02·14-s + 0.750·15-s − 0.625·16-s + 0.174·17-s − 0.280·18-s − 1.20·19-s − 0.376·20-s + 0.699·21-s − 0.254·22-s + 0.379·23-s + 0.627·24-s + 0.688·25-s − 1.05·26-s + 0.192·27-s − 0.351·28-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

\(L(2)\) \(\approx\) \(2.938290409\)
\(L(\frac12)\) \(\approx\) \(2.938290409\)
\(L(\frac{5}{2})\) 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 \( 1 + 2.38T + 8T^{2} \)
5 \( 1 - 14.5T + 125T^{2} \)
7 \( 1 - 22.4T + 343T^{2} \)
13 \( 1 - 58.9T + 2.19e3T^{2} \)
17 \( 1 - 12.2T + 4.91e3T^{2} \)
19 \( 1 + 99.5T + 6.85e3T^{2} \)
23 \( 1 - 41.8T + 1.21e4T^{2} \)
29 \( 1 - 47.8T + 2.43e4T^{2} \)
31 \( 1 - 1.36T + 2.97e4T^{2} \)
37 \( 1 - 405.T + 5.06e4T^{2} \)
41 \( 1 - 248.T + 6.89e4T^{2} \)
43 \( 1 - 188.T + 7.95e4T^{2} \)
47 \( 1 + 417.T + 1.03e5T^{2} \)
53 \( 1 - 52.9T + 1.48e5T^{2} \)
59 \( 1 - 621.T + 2.05e5T^{2} \)
67 \( 1 - 749.T + 3.00e5T^{2} \)
71 \( 1 + 351.T + 3.57e5T^{2} \)
73 \( 1 + 243.T + 3.89e5T^{2} \)
79 \( 1 + 820.T + 4.93e5T^{2} \)
83 \( 1 - 449.T + 5.71e5T^{2} \)
89 \( 1 + 1.26e3T + 7.04e5T^{2} \)
97 \( 1 - 985.T + 9.12e5T^{2} \)
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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

−8.628071373901356043235113584479, −8.416973305320083809054064495267, −7.54981800613015870660268671248, −6.47972748864439319441105209403, −5.68099737957001659129419989366, −4.67273394314337846032073420031, −3.97108405592354180534018100330, −2.46921714181357104954406686259, −1.61834954732477811847931463857, −0.989802647859029715807972427924, 0.989802647859029715807972427924, 1.61834954732477811847931463857, 2.46921714181357104954406686259, 3.97108405592354180534018100330, 4.67273394314337846032073420031, 5.68099737957001659129419989366, 6.47972748864439319441105209403, 7.54981800613015870660268671248, 8.416973305320083809054064495267, 8.628071373901356043235113584479

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