Properties

Label 2-2013-1.1-c1-0-91
Degree $2$
Conductor $2013$
Sign $-1$
Analytic cond. $16.0738$
Root an. cond. $4.00922$
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  + 1.65·2-s − 3-s + 0.731·4-s + 2.07·5-s − 1.65·6-s − 0.262·7-s − 2.09·8-s + 9-s + 3.42·10-s + 11-s − 0.731·12-s − 6.76·13-s − 0.433·14-s − 2.07·15-s − 4.92·16-s − 4.89·17-s + 1.65·18-s + 0.504·19-s + 1.51·20-s + 0.262·21-s + 1.65·22-s − 5.31·23-s + 2.09·24-s − 0.692·25-s − 11.1·26-s − 27-s − 0.191·28-s + ⋯
L(s)  = 1  + 1.16·2-s − 0.577·3-s + 0.365·4-s + 0.928·5-s − 0.674·6-s − 0.0992·7-s − 0.741·8-s + 0.333·9-s + 1.08·10-s + 0.301·11-s − 0.211·12-s − 1.87·13-s − 0.115·14-s − 0.535·15-s − 1.23·16-s − 1.18·17-s + 0.389·18-s + 0.115·19-s + 0.339·20-s + 0.0572·21-s + 0.352·22-s − 1.10·23-s + 0.428·24-s − 0.138·25-s − 2.19·26-s − 0.192·27-s − 0.0362·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $-1$
Analytic conductor: \(16.0738\)
Root analytic conductor: \(4.00922\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2013,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 + T \)
11 \( 1 - T \)
61 \( 1 - T \)
good2 \( 1 - 1.65T + 2T^{2} \)
5 \( 1 - 2.07T + 5T^{2} \)
7 \( 1 + 0.262T + 7T^{2} \)
13 \( 1 + 6.76T + 13T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 - 0.504T + 19T^{2} \)
23 \( 1 + 5.31T + 23T^{2} \)
29 \( 1 - 2.40T + 29T^{2} \)
31 \( 1 + 0.494T + 31T^{2} \)
37 \( 1 - 5.39T + 37T^{2} \)
41 \( 1 + 1.34T + 41T^{2} \)
43 \( 1 - 0.0199T + 43T^{2} \)
47 \( 1 - 3.84T + 47T^{2} \)
53 \( 1 + 1.86T + 53T^{2} \)
59 \( 1 + 6.37T + 59T^{2} \)
67 \( 1 + 0.908T + 67T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 + 0.205T + 79T^{2} \)
83 \( 1 + 8.60T + 83T^{2} \)
89 \( 1 - 4.11T + 89T^{2} \)
97 \( 1 + 14.6T + 97T^{2} \)
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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

−9.001305563429883553532430175437, −7.76791037922175545449856316461, −6.75611364265672681240255006879, −6.20701871474385249848725139050, −5.45975257280332656514580695039, −4.73890673643414475886062002442, −4.12842501238687471909044173817, −2.78789312824455701287437423431, −1.99950814378009503297542023261, 0, 1.99950814378009503297542023261, 2.78789312824455701287437423431, 4.12842501238687471909044173817, 4.73890673643414475886062002442, 5.45975257280332656514580695039, 6.20701871474385249848725139050, 6.75611364265672681240255006879, 7.76791037922175545449856316461, 9.001305563429883553532430175437

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