Properties

Label 1-927-927.317-r1-0-0
Degree $1$
Conductor $927$
Sign $-0.936 + 0.349i$
Analytic cond. $99.6199$
Root an. cond. $99.6199$
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  + (0.779 + 0.626i)2-s + (0.213 + 0.976i)4-s + (0.696 + 0.717i)5-s + (0.552 − 0.833i)7-s + (−0.445 + 0.895i)8-s + (0.0922 + 0.995i)10-s + (0.779 + 0.626i)11-s + (−0.998 − 0.0615i)13-s + (0.952 − 0.303i)14-s + (−0.908 + 0.417i)16-s + (0.850 + 0.526i)17-s + (−0.982 − 0.183i)19-s + (−0.552 + 0.833i)20-s + (0.213 + 0.976i)22-s + (0.779 − 0.626i)23-s + ⋯
L(s)  = 1  + (0.779 + 0.626i)2-s + (0.213 + 0.976i)4-s + (0.696 + 0.717i)5-s + (0.552 − 0.833i)7-s + (−0.445 + 0.895i)8-s + (0.0922 + 0.995i)10-s + (0.779 + 0.626i)11-s + (−0.998 − 0.0615i)13-s + (0.952 − 0.303i)14-s + (−0.908 + 0.417i)16-s + (0.850 + 0.526i)17-s + (−0.982 − 0.183i)19-s + (−0.552 + 0.833i)20-s + (0.213 + 0.976i)22-s + (0.779 − 0.626i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.936 + 0.349i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ -0.936 + 0.349i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6376599431 + 3.530445410i\)
\(L(\frac12)\) \(\approx\) \(0.6376599431 + 3.530445410i\)
\(L(1)\) \(\approx\) \(1.461882618 + 1.133874341i\)
\(L(1)\) \(\approx\) \(1.461882618 + 1.133874341i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.779 + 0.626i)T \)
5 \( 1 + (0.696 + 0.717i)T \)
7 \( 1 + (0.552 - 0.833i)T \)
11 \( 1 + (0.779 + 0.626i)T \)
13 \( 1 + (-0.998 - 0.0615i)T \)
17 \( 1 + (0.850 + 0.526i)T \)
19 \( 1 + (-0.982 - 0.183i)T \)
23 \( 1 + (0.779 - 0.626i)T \)
29 \( 1 + (-0.969 + 0.243i)T \)
31 \( 1 + (0.816 + 0.577i)T \)
37 \( 1 + (-0.602 + 0.798i)T \)
41 \( 1 + (-0.969 - 0.243i)T \)
43 \( 1 + (-0.389 + 0.920i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.982 + 0.183i)T \)
59 \( 1 + (-0.552 - 0.833i)T \)
61 \( 1 + (-0.0307 + 0.999i)T \)
67 \( 1 + (0.552 + 0.833i)T \)
71 \( 1 + (0.273 + 0.961i)T \)
73 \( 1 + (-0.273 - 0.961i)T \)
79 \( 1 + (0.969 - 0.243i)T \)
83 \( 1 + (-0.552 + 0.833i)T \)
89 \( 1 + (-0.739 - 0.673i)T \)
97 \( 1 + (0.881 + 0.473i)T \)
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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

−21.371487968268494469607401993582, −20.81282668861858273699163426541, −19.86918320299010711461542801930, −19.058558804197580733530131176367, −18.43244082255326799791180087793, −17.1817146297030095012096634930, −16.72960974919679321681174254531, −15.43119523625982322705192190869, −14.78521582252846221137827621681, −14.007528841980473978755933873178, −13.304903952773825563799654538153, −12.26246931504421846700768482001, −11.945363477922837077046527489573, −10.97235832610311770851364491797, −9.88329041993913901342740741407, −9.24951863535110273885367415519, −8.43696203912842778598454960628, −7.044635202869211181891425838640, −5.91382960798513237067951165592, −5.37127648017944698286252431105, −4.6179349649137660555190603094, −3.4979061649863284542199231894, −2.33690595451185878814468354282, −1.67242644912332247962800117928, −0.52383673675717082397758148303, 1.478166700421928989342016653767, 2.51853125447811275368601918363, 3.56059720585112846301957667748, 4.51653300513062296495084580581, 5.25965927955939504505050309714, 6.44106533446017797620712320493, 6.95643587009585199071453131417, 7.72151696868009936599352506016, 8.79087519191668529773283971678, 9.9567856325069956646234274335, 10.68437759297128314780282315423, 11.67589914066022623686519879447, 12.55988034751426054663579865462, 13.35561057802850623372138999624, 14.30644885881012569792160538216, 14.66556685503450444042360404338, 15.25212574486066196073567317569, 16.81959547131076958071866497222, 17.06224092946477447910448665997, 17.66219434825617493294939265459, 18.79878016816990282051599458834, 19.79198798193569267725782371934, 20.74955090160813419935394355555, 21.33897515757892702909433650774, 22.16794605275736874834061382862

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