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Properties

Label 2-2013-2013.731-c0-0-7
Degree $2$
Conductor $2013$
Sign $0.111 + 0.993i$
Analytic cond. $1.00461$
Root an. cond. $1.00230$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.280 − 0.863i)2-s + (0.809 + 0.587i)3-s + (0.142 − 0.103i)4-s + (0.280 − 0.863i)6-s + (−0.863 − 0.627i)8-s + (0.309 + 0.951i)9-s + (0.156 − 0.987i)11-s + 0.175·12-s + (−0.587 − 1.80i)13-s + (−0.245 + 0.754i)16-s + (0.0966 − 0.297i)17-s + (0.734 − 0.533i)18-s + (0.951 + 0.690i)19-s + (−0.896 + 0.142i)22-s + 1.78·23-s + (−0.329 − 1.01i)24-s + ⋯
L(s)  = 1  + (−0.280 − 0.863i)2-s + (0.809 + 0.587i)3-s + (0.142 − 0.103i)4-s + (0.280 − 0.863i)6-s + (−0.863 − 0.627i)8-s + (0.309 + 0.951i)9-s + (0.156 − 0.987i)11-s + 0.175·12-s + (−0.587 − 1.80i)13-s + (−0.245 + 0.754i)16-s + (0.0966 − 0.297i)17-s + (0.734 − 0.533i)18-s + (0.951 + 0.690i)19-s + (−0.896 + 0.142i)22-s + 1.78·23-s + (−0.329 − 1.01i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $0.111 + 0.993i$
Analytic conductor: \(1.00461\)
Root analytic conductor: \(1.00230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2013} (731, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2013,\ (\ :0),\ 0.111 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.402077507\)
\(L(\frac12)\) \(\approx\) \(1.402077507\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.156 + 0.987i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
good2 \( 1 + (0.280 + 0.863i)T + (-0.809 + 0.587i)T^{2} \)
5 \( 1 + (0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.0966 + 0.297i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 - 1.78T + T^{2} \)
29 \( 1 + (1.44 - 1.04i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.280 - 0.863i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-1.59 + 1.16i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.437 - 1.34i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 + 0.312T + T^{2} \)
97 \( 1 + (-0.809 + 0.587i)T^{2} \)
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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

−9.315035338560960256554389070002, −8.628012236215063340008559681234, −7.78135831325458974941872759336, −7.04905047610319329089468723479, −5.65126065012334663037306626963, −5.24766800427858436336158452303, −3.67909422450789072868921422295, −3.19957424010240290410615521056, −2.45077683731948893226885205996, −1.06291755426507136053271374538, 1.74917875758040828638643405244, 2.55261116753238800842275514719, 3.68129814404468257338792551114, 4.72189670813235827187011713444, 5.83410518816620507164410048103, 6.88770659323735913220479454786, 7.14321676497234726020557651502, 7.68359235184045049626745202976, 8.712969167265629486573860247113, 9.387935750433285289010490160117

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