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Properties

Label 1-2013-2013.104-r0-0-0
Degree $1$
Conductor $2013$
Sign $-0.629 - 0.777i$
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  + (−0.743 − 0.669i)2-s + (0.104 + 0.994i)4-s + (0.669 − 0.743i)5-s + (0.866 − 0.5i)7-s + (0.587 − 0.809i)8-s + (−0.994 + 0.104i)10-s + (0.913 + 0.406i)13-s + (−0.978 − 0.207i)14-s + (−0.978 + 0.207i)16-s + (0.207 − 0.978i)17-s + (0.5 − 0.866i)19-s + (0.809 + 0.587i)20-s + (−0.587 + 0.809i)23-s + (−0.104 − 0.994i)25-s + (−0.406 − 0.913i)26-s + ⋯
L(s)  = 1  + (−0.743 − 0.669i)2-s + (0.104 + 0.994i)4-s + (0.669 − 0.743i)5-s + (0.866 − 0.5i)7-s + (0.587 − 0.809i)8-s + (−0.994 + 0.104i)10-s + (0.913 + 0.406i)13-s + (−0.978 − 0.207i)14-s + (−0.978 + 0.207i)16-s + (0.207 − 0.978i)17-s + (0.5 − 0.866i)19-s + (0.809 + 0.587i)20-s + (−0.587 + 0.809i)23-s + (−0.104 − 0.994i)25-s + (−0.406 − 0.913i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $-0.629 - 0.777i$
Analytic conductor: \(9.34833\)
Root analytic conductor: \(9.34833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2013} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2013,\ (0:\ ),\ -0.629 - 0.777i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5950026774 - 1.247400060i\)
\(L(\frac12)\) \(\approx\) \(0.5950026774 - 1.247400060i\)
\(L(1)\) \(\approx\) \(0.8126601632 - 0.5083188060i\)
\(L(1)\) \(\approx\) \(0.8126601632 - 0.5083188060i\)

Euler product

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

−20.270433643395712867642749083292, −19.02398366312846684021635318582, −18.560265709101207652572714220, −18.08922911664858283617743029006, −17.35527470123145444202241332890, −16.76718473318312695395976970424, −15.72069473241789613074740820701, −15.18996845006161368018521557575, −14.357656005210157860131966235686, −14.0557059530028503698447042655, −12.99303403699898942447173178154, −11.8933300169895881504434830221, −10.99689527673127244198408502674, −10.48530120867741575388505695564, −9.8020823878402714803203326965, −8.831312618709855873770968500, −8.199945215870632533199395893602, −7.5523614390342514725213566133, −6.49655572761936982517951822744, −5.880500264101293965905400637951, −5.37559567780113259089943671958, −4.16412393199780667711126889321, −2.96759240547185151044244180889, −1.8552912341344178829260195944, −1.356111421689271327664942252330, 0.62677866837527172972003945943, 1.50830827176204725094447784300, 2.11483514238610012093125487131, 3.324598503500697174800288397940, 4.22097464162394367580555993443, 5.0132311843695918193338098382, 5.94655951770127414149199449256, 7.188749342834780927968832330608, 7.702031329585712146117928979404, 8.72742674658477896460094275194, 9.189479625335987102691252741653, 9.90036264321279726933824164028, 10.86049481023056441036444172924, 11.41002061882703938538521795880, 12.0930627492088761385180978984, 13.12257568954813430934474444962, 13.62124817886009439000510339078, 14.25872596323247448994551225408, 15.62772506793218135195153954287, 16.26229037519876663268610094048, 16.963302178470989495406613669047, 17.6167613542210811042467539906, 18.1646577327041514619462855436, 18.80591714349358037569815255555, 19.93005848790478145558926639427

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