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Properties

Label 2-2013-2013.1829-c0-0-5
Degree $2$
Conductor $2013$
Sign $0.132 + 0.991i$
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  + (−0.253 + 0.183i)2-s + (−0.309 − 0.951i)3-s + (−0.278 + 0.857i)4-s + (0.253 + 0.183i)6-s + (−0.183 − 0.565i)8-s + (−0.809 + 0.587i)9-s + (0.891 − 0.453i)11-s + 0.902·12-s + (−0.951 + 0.690i)13-s + (−0.579 − 0.420i)16-s + (−1.44 − 1.04i)17-s + (0.0966 − 0.297i)18-s + (−0.587 − 1.80i)19-s + (−0.142 + 0.278i)22-s + 1.97·23-s + (−0.481 + 0.349i)24-s + ⋯
L(s)  = 1  + (−0.253 + 0.183i)2-s + (−0.309 − 0.951i)3-s + (−0.278 + 0.857i)4-s + (0.253 + 0.183i)6-s + (−0.183 − 0.565i)8-s + (−0.809 + 0.587i)9-s + (0.891 − 0.453i)11-s + 0.902·12-s + (−0.951 + 0.690i)13-s + (−0.579 − 0.420i)16-s + (−1.44 − 1.04i)17-s + (0.0966 − 0.297i)18-s + (−0.587 − 1.80i)19-s + (−0.142 + 0.278i)22-s + 1.97·23-s + (−0.481 + 0.349i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $0.132 + 0.991i$
Analytic conductor: \(1.00461\)
Root analytic conductor: \(1.00230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2013} (1829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2013,\ (\ :0),\ 0.132 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6401270430\)
\(L(\frac12)\) \(\approx\) \(0.6401270430\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.891 + 0.453i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (0.253 - 0.183i)T + (0.309 - 0.951i)T^{2} \)
5 \( 1 + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (1.44 + 1.04i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 - 1.97T + T^{2} \)
29 \( 1 + (-0.610 + 1.87i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.253 + 0.183i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.280 + 0.863i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (1.14 + 0.831i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + 1.78T + T^{2} \)
97 \( 1 + (0.309 - 0.951i)T^{2} \)
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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

−9.044173460191189028977452049188, −8.472359349387517411957225553174, −7.34931737612649436787281311152, −6.89471228331741821341817044896, −6.50373770155722588495557965572, −4.97507025416204347255522652891, −4.47480499813192804503055738041, −3.02427361495725576913069974124, −2.30387307424830489558158830273, −0.55111017882565475158532499638, 1.38384935506887729006728618218, 2.71329779071612337336297597971, 3.98483119329863074498391454149, 4.68100889754307757107757453313, 5.39100567419820356598686396406, 6.28855617928348172967686299648, 6.92813774054187611315026778620, 8.486263720969424226364050739596, 8.775388219035362718681589322940, 9.670578938830417391051564856254

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