Properties

Label 2-2013-1.1-c3-0-106
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.42·2-s − 3·3-s + 3.72·4-s − 0.330·5-s + 10.2·6-s − 34.8·7-s + 14.6·8-s + 9·9-s + 1.13·10-s − 11·11-s − 11.1·12-s + 25.0·13-s + 119.·14-s + 0.990·15-s − 79.9·16-s − 71.1·17-s − 30.8·18-s + 74.1·19-s − 1.23·20-s + 104.·21-s + 37.6·22-s + 13.0·23-s − 43.8·24-s − 124.·25-s − 85.7·26-s − 27·27-s − 130.·28-s + ⋯
L(s)  = 1  − 1.21·2-s − 0.577·3-s + 0.466·4-s − 0.0295·5-s + 0.699·6-s − 1.88·7-s + 0.646·8-s + 0.333·9-s + 0.0357·10-s − 0.301·11-s − 0.269·12-s + 0.534·13-s + 2.27·14-s + 0.0170·15-s − 1.24·16-s − 1.01·17-s − 0.403·18-s + 0.895·19-s − 0.0137·20-s + 1.08·21-s + 0.365·22-s + 0.117·23-s − 0.373·24-s − 0.999·25-s − 0.647·26-s − 0.192·27-s − 0.877·28-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: \(1\)
Selberg data: \((2,\ 2013,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.42T + 8T^{2} \)
5 \( 1 + 0.330T + 125T^{2} \)
7 \( 1 + 34.8T + 343T^{2} \)
13 \( 1 - 25.0T + 2.19e3T^{2} \)
17 \( 1 + 71.1T + 4.91e3T^{2} \)
19 \( 1 - 74.1T + 6.85e3T^{2} \)
23 \( 1 - 13.0T + 1.21e4T^{2} \)
29 \( 1 + 107.T + 2.43e4T^{2} \)
31 \( 1 - 21.5T + 2.97e4T^{2} \)
37 \( 1 + 32.4T + 5.06e4T^{2} \)
41 \( 1 - 11.4T + 6.89e4T^{2} \)
43 \( 1 - 67.7T + 7.95e4T^{2} \)
47 \( 1 + 283.T + 1.03e5T^{2} \)
53 \( 1 - 234.T + 1.48e5T^{2} \)
59 \( 1 - 315.T + 2.05e5T^{2} \)
67 \( 1 + 902.T + 3.00e5T^{2} \)
71 \( 1 - 742.T + 3.57e5T^{2} \)
73 \( 1 - 531.T + 3.89e5T^{2} \)
79 \( 1 - 1.35e3T + 4.93e5T^{2} \)
83 \( 1 + 1.44e3T + 5.71e5T^{2} \)
89 \( 1 - 495.T + 7.04e5T^{2} \)
97 \( 1 - 605.T + 9.12e5T^{2} \)
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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

−8.601117620805056390712187984724, −7.62129537969286697841465014344, −6.91768835792201796290064241342, −6.27760163357674173043600036762, −5.41578218735076917822319617304, −4.20720952097112046725708598268, −3.32030265253004746466443116728, −2.08847827214795889686338457094, −0.75323184643022000560089014451, 0, 0.75323184643022000560089014451, 2.08847827214795889686338457094, 3.32030265253004746466443116728, 4.20720952097112046725708598268, 5.41578218735076917822319617304, 6.27760163357674173043600036762, 6.91768835792201796290064241342, 7.62129537969286697841465014344, 8.601117620805056390712187984724

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