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Properties

Label 1-2340-2340.2047-r1-0-0
Degree $1$
Conductor $2340$
Sign $0.771 + 0.636i$
Analytic cond. $251.467$
Root an. cond. $251.467$
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  + (−0.5 − 0.866i)7-s − i·11-s + (0.866 + 0.5i)17-s + (0.866 + 0.5i)19-s + (0.866 + 0.5i)23-s − 29-s + (0.866 − 0.5i)31-s + (0.5 + 0.866i)37-s + (0.866 + 0.5i)41-s + (−0.866 + 0.5i)43-s + (−0.5 + 0.866i)47-s + (−0.5 + 0.866i)49-s − i·53-s − i·59-s + (−0.5 − 0.866i)61-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s − i·11-s + (0.866 + 0.5i)17-s + (0.866 + 0.5i)19-s + (0.866 + 0.5i)23-s − 29-s + (0.866 − 0.5i)31-s + (0.5 + 0.866i)37-s + (0.866 + 0.5i)41-s + (−0.866 + 0.5i)43-s + (−0.5 + 0.866i)47-s + (−0.5 + 0.866i)49-s − i·53-s − i·59-s + (−0.5 − 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.771 + 0.636i$
Analytic conductor: \(251.467\)
Root analytic conductor: \(251.467\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (2047, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2340,\ (1:\ ),\ 0.771 + 0.636i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.734050419 + 0.6226743196i\)
\(L(\frac12)\) \(\approx\) \(1.734050419 + 0.6226743196i\)
\(L(1)\) \(\approx\) \(1.052798504 - 0.05183198057i\)
\(L(1)\) \(\approx\) \(1.052798504 - 0.05183198057i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 - iT \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 - T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 - iT \)
59 \( 1 - iT \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
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

−19.251793198191228057244633097888, −18.6149187564933181448785742896, −18.01845396426771603024156104093, −17.23435644399121406197579628053, −16.38654939785809366749787632691, −15.75552817196217606548957642621, −15.041683033963963051029991404813, −14.4545721899259405849952440684, −13.4782539051126805436190273267, −12.74043646319645562049264009010, −12.12812813311054343803504927245, −11.50279591654433121560481863636, −10.4994078238264099908108931670, −9.6387238025306923070650455686, −9.24261863873494944573935229016, −8.32497087820544401483608892431, −7.32139963606574472702562334513, −6.843659012051775860915884121615, −5.71586782260964468653863639399, −5.19276831256545628435668371762, −4.26064726092394141427813357670, −3.15478530169999043044946525051, −2.56655466649472758496216334046, −1.53146183305465019662623069091, −0.39212662690182848280986978270, 0.81437220626021678007520189397, 1.42345960749579666968231655531, 2.96007177417634694515631343343, 3.40867508915500454183347294660, 4.28470130331139571273837824122, 5.33982710215247463798424124060, 6.061744612988623825219890825376, 6.829425076005108608302198731718, 7.7890073193611948019718878008, 8.2131038851427650648394311543, 9.51834098933350687466104481283, 9.79518277416658008096039622471, 10.866744442154657029118058204899, 11.34024594639638026851096704306, 12.29337480610989647873542750922, 13.16430542418693734123316929075, 13.62993616435147013663361830840, 14.40368007040768135081149984177, 15.16001501569499515291542249567, 16.114212844796796259772978378, 16.67792897301045742200451274762, 17.11390060040433003611889521038, 18.16281456655795612715046916698, 18.895247566286547400548202822294, 19.43477497818076790689362464784

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