Properties

Label 2-2013-1.1-c1-0-74
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  + 0.423·2-s + 3-s − 1.82·4-s − 1.36·5-s + 0.423·6-s + 0.865·7-s − 1.61·8-s + 9-s − 0.577·10-s − 11-s − 1.82·12-s + 2.63·13-s + 0.366·14-s − 1.36·15-s + 2.95·16-s − 3.12·17-s + 0.423·18-s − 0.281·19-s + 2.48·20-s + 0.865·21-s − 0.423·22-s + 1.73·23-s − 1.61·24-s − 3.13·25-s + 1.11·26-s + 27-s − 1.57·28-s + ⋯
L(s)  = 1  + 0.299·2-s + 0.577·3-s − 0.910·4-s − 0.610·5-s + 0.172·6-s + 0.327·7-s − 0.571·8-s + 0.333·9-s − 0.182·10-s − 0.301·11-s − 0.525·12-s + 0.731·13-s + 0.0978·14-s − 0.352·15-s + 0.739·16-s − 0.758·17-s + 0.0997·18-s − 0.0646·19-s + 0.555·20-s + 0.188·21-s − 0.0902·22-s + 0.362·23-s − 0.329·24-s − 0.627·25-s + 0.218·26-s + 0.192·27-s − 0.297·28-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 \( 1 - 0.423T + 2T^{2} \)
5 \( 1 + 1.36T + 5T^{2} \)
7 \( 1 - 0.865T + 7T^{2} \)
13 \( 1 - 2.63T + 13T^{2} \)
17 \( 1 + 3.12T + 17T^{2} \)
19 \( 1 + 0.281T + 19T^{2} \)
23 \( 1 - 1.73T + 23T^{2} \)
29 \( 1 + 7.07T + 29T^{2} \)
31 \( 1 + 4.28T + 31T^{2} \)
37 \( 1 - 3.94T + 37T^{2} \)
41 \( 1 + 7.38T + 41T^{2} \)
43 \( 1 - 5.77T + 43T^{2} \)
47 \( 1 + 3.73T + 47T^{2} \)
53 \( 1 + 5.86T + 53T^{2} \)
59 \( 1 + 7.96T + 59T^{2} \)
67 \( 1 - 4.80T + 67T^{2} \)
71 \( 1 + 7.35T + 71T^{2} \)
73 \( 1 + 16.3T + 73T^{2} \)
79 \( 1 + 6.03T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−8.801182196745599719492747932568, −8.025315504351552006939727536747, −7.47561088498641935933059723427, −6.31060224618463206245081864428, −5.39335777798924193731996540751, −4.49241596427811710014009028096, −3.85723143127192489627914613661, −3.05943028939640378825534977570, −1.65634094120654582061591255767, 0, 1.65634094120654582061591255767, 3.05943028939640378825534977570, 3.85723143127192489627914613661, 4.49241596427811710014009028096, 5.39335777798924193731996540751, 6.31060224618463206245081864428, 7.47561088498641935933059723427, 8.025315504351552006939727536747, 8.801182196745599719492747932568

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