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Properties

Label 1-663-663.200-r0-0-0
Degree $1$
Conductor $663$
Sign $-0.360 - 0.932i$
Analytic cond. $3.07895$
Root an. cond. $3.07895$
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  − i·2-s − 4-s − 5-s + 7-s + i·8-s + i·10-s − 11-s − i·14-s + 16-s + i·19-s + 20-s + i·22-s − i·23-s + 25-s − 28-s − i·29-s + ⋯
L(s)  = 1  − i·2-s − 4-s − 5-s + 7-s + i·8-s + i·10-s − 11-s − i·14-s + 16-s + i·19-s + 20-s + i·22-s − i·23-s + 25-s − 28-s − i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(663\)    =    \(3 \cdot 13 \cdot 17\)
Sign: $-0.360 - 0.932i$
Analytic conductor: \(3.07895\)
Root analytic conductor: \(3.07895\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{663} (200, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 663,\ (0:\ ),\ -0.360 - 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5434094560 - 0.7926919535i\)
\(L(\frac12)\) \(\approx\) \(0.5434094560 - 0.7926919535i\)
\(L(1)\) \(\approx\) \(0.7277070457 - 0.4412961150i\)
\(L(1)\) \(\approx\) \(0.7277070457 - 0.4412961150i\)

Euler product

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

−23.33365983702561582215842383491, −22.50455829826235073601542863259, −21.50313284911740481211518981114, −20.71829676777437117505942945970, −19.57180038603555409347620670604, −18.85337597864263979846368342777, −17.86735724023192935280394396294, −17.4341586983194023092677815402, −16.17122737648658602245182705471, −15.65103184965725346021269506021, −14.96956237225984038475234691954, −14.11326775180142637828901377214, −13.21575821773731888853617953467, −12.2742068780983662781088704296, −11.259721658458254769571228496611, −10.436237827707128068785710584238, −9.102683414926896142423108831356, −8.333852235742160284364129939949, −7.598182603763783030284947120458, −7.01155197648566210875638802943, −5.58791965085786549365278117305, −4.86172980097771225306270400586, −4.06967954689886066806520499354, −2.81829359026213437537272708224, −1.014438026012325277048935451853, 0.61746547167455524904543973471, 1.9797980196139860341693635144, 2.96854646978589449938593796600, 4.125652861416185951842521701047, 4.725986334017340690010972203530, 5.785081551446155807356912392185, 7.4941142059723920710075750942, 8.13939763852917748182007563927, 8.81302550902064508710317452247, 10.25947252855598180109309318131, 10.714863230855555461857478920578, 11.73461446787490959563171119146, 12.201461768270791396504008648486, 13.20045130379471310811495031621, 14.18934968191630862473184651489, 14.92796486749394424139320704377, 15.87055706949818880365213854965, 16.964291193858436058339930572897, 17.89944837007234610678858300670, 18.661006104233903836114515951488, 19.21863470246612562479995522210, 20.29463807410398132063234153829, 20.837767479889117477865564914079, 21.41929264901304983603551303948, 22.76624931759189045852402751077

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