Properties

Label 2-2013-1.1-c1-0-81
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  + 2.62·2-s + 3-s + 4.88·4-s + 0.233·5-s + 2.62·6-s + 1.03·7-s + 7.57·8-s + 9-s + 0.612·10-s − 11-s + 4.88·12-s + 3.46·13-s + 2.71·14-s + 0.233·15-s + 10.1·16-s + 0.866·17-s + 2.62·18-s − 6.81·19-s + 1.14·20-s + 1.03·21-s − 2.62·22-s − 2.06·23-s + 7.57·24-s − 4.94·25-s + 9.08·26-s + 27-s + 5.06·28-s + ⋯
L(s)  = 1  + 1.85·2-s + 0.577·3-s + 2.44·4-s + 0.104·5-s + 1.07·6-s + 0.391·7-s + 2.67·8-s + 0.333·9-s + 0.193·10-s − 0.301·11-s + 1.41·12-s + 0.960·13-s + 0.726·14-s + 0.0602·15-s + 2.52·16-s + 0.210·17-s + 0.618·18-s − 1.56·19-s + 0.255·20-s + 0.225·21-s − 0.559·22-s − 0.429·23-s + 1.54·24-s − 0.989·25-s + 1.78·26-s + 0.192·27-s + 0.956·28-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

\(L(1)\) \(\approx\) \(7.209332241\)
\(L(\frac12)\) \(\approx\) \(7.209332241\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.62T + 2T^{2} \)
5 \( 1 - 0.233T + 5T^{2} \)
7 \( 1 - 1.03T + 7T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 - 0.866T + 17T^{2} \)
19 \( 1 + 6.81T + 19T^{2} \)
23 \( 1 + 2.06T + 23T^{2} \)
29 \( 1 + 6.06T + 29T^{2} \)
31 \( 1 - 0.640T + 31T^{2} \)
37 \( 1 - 6.49T + 37T^{2} \)
41 \( 1 + 0.461T + 41T^{2} \)
43 \( 1 + 5.84T + 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 - 5.93T + 53T^{2} \)
59 \( 1 + 13.8T + 59T^{2} \)
67 \( 1 - 5.49T + 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 + 15.4T + 73T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 - 4.62T + 83T^{2} \)
89 \( 1 - 4.71T + 89T^{2} \)
97 \( 1 + 0.332T + 97T^{2} \)
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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

−9.053424332221278853822942113984, −8.078076210503489253923413549600, −7.49199773323032067714864972498, −6.38461192078687676945844512057, −5.94940024777582450674828516942, −4.97737749520538628901642449615, −4.11131175534998206412179262288, −3.60093550695772683681744700397, −2.46799645289894480474853416571, −1.73871492647448873279211603254, 1.73871492647448873279211603254, 2.46799645289894480474853416571, 3.60093550695772683681744700397, 4.11131175534998206412179262288, 4.97737749520538628901642449615, 5.94940024777582450674828516942, 6.38461192078687676945844512057, 7.49199773323032067714864972498, 8.078076210503489253923413549600, 9.053424332221278853822942113984

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