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Properties

Label 1-2013-2013.113-r1-0-0
Degree $1$
Conductor $2013$
Sign $0.677 - 0.735i$
Analytic cond. $216.326$
Root an. cond. $216.326$
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  + 2-s + 4-s + (0.809 + 0.587i)5-s + (0.809 − 0.587i)7-s + 8-s + (0.809 + 0.587i)10-s + (0.309 − 0.951i)13-s + (0.809 − 0.587i)14-s + 16-s + (−0.809 − 0.587i)17-s + (−0.809 − 0.587i)19-s + (0.809 + 0.587i)20-s + (−0.809 + 0.587i)23-s + (0.309 + 0.951i)25-s + (0.309 − 0.951i)26-s + ⋯
L(s)  = 1  + 2-s + 4-s + (0.809 + 0.587i)5-s + (0.809 − 0.587i)7-s + 8-s + (0.809 + 0.587i)10-s + (0.309 − 0.951i)13-s + (0.809 − 0.587i)14-s + 16-s + (−0.809 − 0.587i)17-s + (−0.809 − 0.587i)19-s + (0.809 + 0.587i)20-s + (−0.809 + 0.587i)23-s + (0.309 + 0.951i)25-s + (0.309 − 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $0.677 - 0.735i$
Analytic conductor: \(216.326\)
Root analytic conductor: \(216.326\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2013} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2013,\ (1:\ ),\ 0.677 - 0.735i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.930277463 - 2.598698399i\)
\(L(\frac12)\) \(\approx\) \(5.930277463 - 2.598698399i\)
\(L(1)\) \(\approx\) \(2.562605628 - 0.2935267856i\)
\(L(1)\) \(\approx\) \(2.562605628 - 0.2935267856i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
61 \( 1 \)
good2 \( 1 + T \)
5 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + (0.809 + 0.587i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.26354920242445393375074348849, −19.173423777527650473370852158292, −18.438254588629197640639010891679, −17.49841586633605469231855287153, −16.83490449464747062601638391329, −16.190479855081418158398481073159, −15.28666856323342513247286660421, −14.59128808036062112470171654079, −14.019698313063224172893115883298, −13.24523417264509125541288212663, −12.59516485478526995084279451954, −11.88656261669377134565864652121, −11.13616630674696962507887085254, −10.39850592730110258731618468874, −9.38584565323247091737111635029, −8.52319155472269276952651674636, −7.888086201593674179421500271100, −6.532711236884213855577189293622, −6.148875199386825645101722621046, −5.31975360193886189590814292640, −4.462905972127146865937250393846, −4.00946743861594586348644152649, −2.4674952392235010254116114585, −2.01893887594203122415057635267, −1.21394394229640528150662077096, 0.69269226191853156970106268848, 1.89713225159488434938336189025, 2.452565696624776959651117707781, 3.47437585820300013216140120617, 4.32434645265133326760786075128, 5.14394815030318238464637003761, 5.892405690349980412755283115442, 6.63312464988762393326043215788, 7.42162528242804725105681622143, 8.135258271892959754055352876274, 9.34665354084310496780759214382, 10.32534889997084972534331155954, 10.913941862760242325672347774317, 11.37586565610791786806205802854, 12.471668658664622429874706871928, 13.295289237602814224875950533, 13.78170709428767968089100518455, 14.35392677785023740281667438906, 15.235959535042699677538119262827, 15.6551181408724251145053797493, 16.808476572925382196151038633380, 17.53088369902314718070065444153, 17.95105516699312205479560930433, 19.052139844758571074154763985714, 19.95106335643520151593640556940

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