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Properties

Label 1-2013-2013.434-r0-0-0
Degree $1$
Conductor $2013$
Sign $0.193 - 0.981i$
Analytic cond. $9.34833$
Root an. cond. $9.34833$
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  + (−0.207 − 0.978i)2-s + (−0.913 + 0.406i)4-s + (0.913 + 0.406i)5-s + (0.406 − 0.913i)7-s + (0.587 + 0.809i)8-s + (0.207 − 0.978i)10-s + (0.913 + 0.406i)13-s + (−0.978 − 0.207i)14-s + (0.669 − 0.743i)16-s + (−0.994 + 0.104i)17-s + (−0.913 + 0.406i)19-s − 20-s + (0.951 + 0.309i)23-s + (0.669 + 0.743i)25-s + (0.207 − 0.978i)26-s + ⋯
L(s)  = 1  + (−0.207 − 0.978i)2-s + (−0.913 + 0.406i)4-s + (0.913 + 0.406i)5-s + (0.406 − 0.913i)7-s + (0.587 + 0.809i)8-s + (0.207 − 0.978i)10-s + (0.913 + 0.406i)13-s + (−0.978 − 0.207i)14-s + (0.669 − 0.743i)16-s + (−0.994 + 0.104i)17-s + (−0.913 + 0.406i)19-s − 20-s + (0.951 + 0.309i)23-s + (0.669 + 0.743i)25-s + (0.207 − 0.978i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $0.193 - 0.981i$
Analytic conductor: \(9.34833\)
Root analytic conductor: \(9.34833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2013} (434, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2013,\ (0:\ ),\ 0.193 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.315579355 - 1.081885025i\)
\(L(\frac12)\) \(\approx\) \(1.315579355 - 1.081885025i\)
\(L(1)\) \(\approx\) \(1.011813652 - 0.4992846211i\)
\(L(1)\) \(\approx\) \(1.011813652 - 0.4992846211i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (-0.207 - 0.978i)T \)
5 \( 1 + (0.913 + 0.406i)T \)
7 \( 1 + (0.406 - 0.913i)T \)
13 \( 1 + (0.913 + 0.406i)T \)
17 \( 1 + (-0.994 + 0.104i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (0.951 + 0.309i)T \)
29 \( 1 + (0.743 - 0.669i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (0.743 + 0.669i)T \)
47 \( 1 + (0.978 + 0.207i)T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (-0.743 - 0.669i)T \)
67 \( 1 + (-0.207 - 0.978i)T \)
71 \( 1 + (0.743 - 0.669i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.994 + 0.104i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.587 + 0.809i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−20.09715849771691397904617856112, −19.08710547766507358197152241201, −18.3422733029216050691934249782, −17.84908820796577987937486368879, −17.263978501133503759980532728116, −16.476558969787100291093370670707, −15.67543252721134590190640367648, −15.12510469566656074340878851508, −14.36360928820438137037764574378, −13.53099389798786958297054756628, −13.000166592744228156299302217827, −12.255169639011674155623225662061, −10.911309237219461228797375863533, −10.44275333220627352275535813667, −9.19818536556077060376487465799, −8.82923869975725355879922626078, −8.386142838233439173732878530096, −7.14150920557993968184122100784, −6.36530401861337674111050271793, −5.76319371328308841502713044219, −4.997481303412890466863029462291, −4.37118569722914041047472181084, −2.96157142854328228633282327389, −1.91173391518922245595108821327, −0.96330479910644633111543479738, 0.79206608930119785578309425192, 1.79863741090610674878826495919, 2.349269127634113422516999645379, 3.56658451891764291589502093076, 4.16321510030987249137741510914, 5.07747510517142429709938831398, 6.10716567762702535445601022272, 6.9498725055794710789275769719, 7.88844457372274549791838828407, 8.85891730229388903171936946062, 9.351368795169907278732750085031, 10.3851710096750946991987085636, 10.86835881865264822063535112298, 11.26446601439851923835200457884, 12.50638196627615501939093012806, 13.19644689240045898307401678552, 13.79313946645664853020669380593, 14.28869655279542168002448187777, 15.25265173445537670349307994952, 16.459305174572493710102770498371, 17.213537829882760288232689760083, 17.63180443849648612809298637223, 18.347034860843850466213290465962, 19.11275958226226668624261393703, 19.74353583747803115726094901813

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