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Properties

Label 1-2013-2013.1241-r0-0-0
Degree $1$
Conductor $2013$
Sign $0.713 + 0.700i$
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  + (0.994 + 0.104i)2-s + (0.978 + 0.207i)4-s + (0.913 + 0.406i)5-s + (0.743 + 0.669i)7-s + (0.951 + 0.309i)8-s + (0.866 + 0.5i)10-s + (−0.104 + 0.994i)13-s + (0.669 + 0.743i)14-s + (0.913 + 0.406i)16-s + (0.994 − 0.104i)17-s + (−0.669 − 0.743i)19-s + (0.809 + 0.587i)20-s − i·23-s + (0.669 + 0.743i)25-s + (−0.207 + 0.978i)26-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)2-s + (0.978 + 0.207i)4-s + (0.913 + 0.406i)5-s + (0.743 + 0.669i)7-s + (0.951 + 0.309i)8-s + (0.866 + 0.5i)10-s + (−0.104 + 0.994i)13-s + (0.669 + 0.743i)14-s + (0.913 + 0.406i)16-s + (0.994 − 0.104i)17-s + (−0.669 − 0.743i)19-s + (0.809 + 0.587i)20-s − i·23-s + (0.669 + 0.743i)25-s + (−0.207 + 0.978i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $0.713 + 0.700i$
Analytic conductor: \(9.34833\)
Root analytic conductor: \(9.34833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2013} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2013,\ (0:\ ),\ 0.713 + 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.197248571 + 1.716949403i\)
\(L(\frac12)\) \(\approx\) \(4.197248571 + 1.716949403i\)
\(L(1)\) \(\approx\) \(2.486295475 + 0.5840446741i\)
\(L(1)\) \(\approx\) \(2.486295475 + 0.5840446741i\)

Euler product

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

−20.1641423059849500826707954074, −19.41618708585601599610995734251, −18.2619830085778756263011072237, −17.57597978211275685611884075992, −16.79529932563802813667894303025, −16.30577903697080833079827271632, −15.23275492272976264761177506869, −14.361130799447844004010655780678, −14.21999254042996172067358777103, −13.021013206675680372256098567031, −12.86111671783783441522601822125, −11.8509290556335501255416571058, −10.99104590267318886705468409983, −10.26562584994256214515117359409, −9.773543252771645464447939135425, −8.38027337180744008731528746678, −7.76312379730581337037611459654, −6.83318595565262850759751334003, −5.910400558415861331771499258465, −5.28162167801174420305350423751, −4.68851619310626887438888271469, −3.63305626905656032688611035239, −2.8846072517829549130066642049, −1.63703298358290965088559735649, −1.25207893166492361914185556466, 1.4273970796160292541881729908, 2.30435297202541724768845385996, 2.7227531877418840662394967071, 4.04572003704265632128581238827, 4.75080427683186286539311348916, 5.60117803794475167509015541190, 6.19578881340386829181171253358, 6.95026201553568816623893590865, 7.84457405409288693589682672449, 8.78438860413960523964229236440, 9.69009132088853950248430039024, 10.56884338115362948032796818554, 11.35443845762599104416814949747, 11.90922282188961718343650869280, 12.81044554464910983164004068588, 13.49519731830932836311567436555, 14.321805081725441012767244266930, 14.68542940439163820291433793504, 15.34419923996647730184706593586, 16.441169440396642391633736913927, 16.95354622416333699592078547461, 17.7807300550408743981368818414, 18.65925614544763213048111915539, 19.21518002379459596857213075242, 20.378477391556059819606655108562

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