Properties

Label 2-2013-1.1-c1-0-6
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.59·2-s − 3-s + 4.75·4-s − 3.50·5-s + 2.59·6-s + 0.252·7-s − 7.15·8-s + 9-s + 9.12·10-s + 11-s − 4.75·12-s + 0.314·13-s − 0.655·14-s + 3.50·15-s + 9.09·16-s + 3.13·17-s − 2.59·18-s + 0.841·19-s − 16.6·20-s − 0.252·21-s − 2.59·22-s + 1.45·23-s + 7.15·24-s + 7.31·25-s − 0.818·26-s − 27-s + 1.19·28-s + ⋯
L(s)  = 1  − 1.83·2-s − 0.577·3-s + 2.37·4-s − 1.56·5-s + 1.06·6-s + 0.0953·7-s − 2.53·8-s + 0.333·9-s + 2.88·10-s + 0.301·11-s − 1.37·12-s + 0.0873·13-s − 0.175·14-s + 0.906·15-s + 2.27·16-s + 0.759·17-s − 0.612·18-s + 0.192·19-s − 3.73·20-s − 0.0550·21-s − 0.554·22-s + 0.302·23-s + 1.46·24-s + 1.46·25-s − 0.160·26-s − 0.192·27-s + 0.226·28-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: \(0\)
Selberg data: \((2,\ 2013,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3082284533\)
\(L(\frac12)\) \(\approx\) \(0.3082284533\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.59T + 2T^{2} \)
5 \( 1 + 3.50T + 5T^{2} \)
7 \( 1 - 0.252T + 7T^{2} \)
13 \( 1 - 0.314T + 13T^{2} \)
17 \( 1 - 3.13T + 17T^{2} \)
19 \( 1 - 0.841T + 19T^{2} \)
23 \( 1 - 1.45T + 23T^{2} \)
29 \( 1 + 6.76T + 29T^{2} \)
31 \( 1 + 7.07T + 31T^{2} \)
37 \( 1 - 8.89T + 37T^{2} \)
41 \( 1 + 8.43T + 41T^{2} \)
43 \( 1 - 1.80T + 43T^{2} \)
47 \( 1 - 2.30T + 47T^{2} \)
53 \( 1 + 4.41T + 53T^{2} \)
59 \( 1 - 4.48T + 59T^{2} \)
67 \( 1 + 0.0414T + 67T^{2} \)
71 \( 1 + 9.80T + 71T^{2} \)
73 \( 1 - 8.90T + 73T^{2} \)
79 \( 1 + 8.51T + 79T^{2} \)
83 \( 1 - 11.2T + 83T^{2} \)
89 \( 1 - 5.58T + 89T^{2} \)
97 \( 1 - 7.54T + 97T^{2} \)
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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

−9.159021112684370405226971268858, −8.281939593239375157619580713115, −7.68847572228506357328687206936, −7.23415062957490701869006800735, −6.40544088829447325074783351353, −5.33613646063702112501229335142, −4.04395412211449067668976821401, −3.13112540859446497145040806868, −1.62557904602805132848864281664, −0.51206171655069171332020761223, 0.51206171655069171332020761223, 1.62557904602805132848864281664, 3.13112540859446497145040806868, 4.04395412211449067668976821401, 5.33613646063702112501229335142, 6.40544088829447325074783351353, 7.23415062957490701869006800735, 7.68847572228506357328687206936, 8.281939593239375157619580713115, 9.159021112684370405226971268858

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