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Properties

Label 1-1860-1860.719-r1-0-0
Degree $1$
Conductor $1860$
Sign $0.695 - 0.718i$
Analytic cond. $199.884$
Root an. cond. $199.884$
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 + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + 23-s + 29-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s − 47-s + (−0.5 − 0.866i)49-s + (0.5 + 0.866i)53-s + (−0.5 + 0.866i)59-s − 61-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + 23-s + 29-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s − 47-s + (−0.5 − 0.866i)49-s + (0.5 + 0.866i)53-s + (−0.5 + 0.866i)59-s − 61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.695 - 0.718i$
Analytic conductor: \(199.884\)
Root analytic conductor: \(199.884\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1860,\ (1:\ ),\ 0.695 - 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.675541574 - 0.7106302259i\)
\(L(\frac12)\) \(\approx\) \(1.675541574 - 0.7106302259i\)
\(L(1)\) \(\approx\) \(1.050567191 + 0.002504622398i\)
\(L(1)\) \(\approx\) \(1.050567191 + 0.002504622398i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
31 \( 1 \)
good7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + T \)
97 \( 1 - T \)
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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

−19.82041990974931305795779576384, −19.28228565439698250556268514536, −18.865357036953937751720871277973, −17.66012072707285160861993931199, −17.01411922664920078702897916895, −16.41029815226112874987293606402, −15.8722047107606142847148923871, −14.4953004529498455536153804497, −14.32240906115175703756871435711, −13.40174422961356616692876087793, −12.62348954922059333818419614470, −11.85693304939945263705493760545, −11.00946491682324451336569127146, −10.29369686520240141431328812171, −9.541260734662542512913755466778, −8.75116757540728800708184227402, −7.85064184316761523498002823410, −7.02190419530564197976687326828, −6.34104574528688321901984342776, −5.50022056933921483911626113747, −4.380927066995859858706893175545, −3.689162683834827443730525593649, −2.93118420770822595058631283602, −1.594967815363517963734784684846, −0.81060821046303437145042637985, 0.413256954499424878007453624793, 1.506139624732085488032195755755, 2.84890000056344316669992581298, 3.01393060363044793342789289997, 4.5617914840288282379500191495, 5.11045241441382821303458114301, 6.030229808750483277550406041454, 6.947042515410092312404975881358, 7.55079863040688108467657146082, 8.642966552395448497038752786231, 9.372502861097701490748464044381, 9.90700989137769810784371913513, 10.84867356839945426064352588029, 11.95986864933634698725612621887, 12.20869504782016882927202090928, 13.12837429574555692601281735768, 13.883061033736419318404928549075, 14.97150913253359789194640991431, 15.26236238484796047654546301042, 16.10722565952436166066653320703, 16.94310950012497270473118994976, 17.76822342865461831699482396467, 18.264766044772174293246053645473, 19.212913362792353358093767603777, 19.79152799890235767742606642881

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