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Properties

Label 1-775-775.133-r1-0-0
Degree $1$
Conductor $775$
Sign $0.887 + 0.461i$
Analytic cond. $83.2853$
Root an. cond. $83.2853$
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 + (0.743 + 0.669i)3-s − 4-s + (0.669 − 0.743i)6-s + (−0.743 − 0.669i)7-s + i·8-s + (0.104 + 0.994i)9-s + (0.913 − 0.406i)11-s + (−0.743 − 0.669i)12-s + (−0.866 − 0.5i)13-s + (−0.669 + 0.743i)14-s + 16-s + (−0.866 + 0.5i)17-s + (0.994 − 0.104i)18-s + (0.978 − 0.207i)19-s + ⋯
L(s)  = 1  − i·2-s + (0.743 + 0.669i)3-s − 4-s + (0.669 − 0.743i)6-s + (−0.743 − 0.669i)7-s + i·8-s + (0.104 + 0.994i)9-s + (0.913 − 0.406i)11-s + (−0.743 − 0.669i)12-s + (−0.866 − 0.5i)13-s + (−0.669 + 0.743i)14-s + 16-s + (−0.866 + 0.5i)17-s + (0.994 − 0.104i)18-s + (0.978 − 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.887 + 0.461i$
Analytic conductor: \(83.2853\)
Root analytic conductor: \(83.2853\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 775,\ (1:\ ),\ 0.887 + 0.461i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.544831835 + 0.3776282894i\)
\(L(\frac12)\) \(\approx\) \(1.544831835 + 0.3776282894i\)
\(L(1)\) \(\approx\) \(1.047035510 - 0.2455711363i\)
\(L(1)\) \(\approx\) \(1.047035510 - 0.2455711363i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 \)
good2 \( 1 - iT \)
3 \( 1 + (0.743 + 0.669i)T \)
7 \( 1 + (-0.743 - 0.669i)T \)
11 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.978 - 0.207i)T \)
23 \( 1 + (-0.951 + 0.309i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.994 + 0.104i)T \)
41 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (-0.994 - 0.104i)T \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (0.406 + 0.913i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.207 + 0.978i)T \)
71 \( 1 + (0.913 - 0.406i)T \)
73 \( 1 + (0.406 + 0.913i)T \)
79 \( 1 + (0.104 - 0.994i)T \)
83 \( 1 + (-0.743 + 0.669i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 - iT \)
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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

−22.213662328485433603118969718711, −21.63185213016700537119994816235, −20.0401812267457271158952404565, −19.72831217411374655920233486410, −18.6304478154202583717698570730, −18.18190222869380398582821375103, −17.2627964174697052250035557107, −16.30157026702942082835463423717, −15.53276568330346896812447561443, −14.70525617012980335335645971960, −14.08079197365342817792190763278, −13.31356966959269842530083579179, −12.33998499267655816222309229134, −11.8615120812631781953888638804, −9.846004786670897153599988631474, −9.40886619824381088027254424076, −8.65108756532503683321673032370, −7.70362868501762003592715474446, −6.735671685355123545307126163019, −6.43027891306923509971482780089, −5.11123184465559750523297043785, −4.03355444741609562674669355026, −3.001745230851597370875988256709, −1.83782031429461728240881285531, −0.38858362391347166524330383813, 0.92258968263124170153096787623, 2.26152166183886799721169600558, 3.16068262253006851624953883993, 3.91967948596378693876911447698, 4.63008701941027948272541783122, 5.86318949995693367473972862737, 7.245450567480111569318044687811, 8.2752133295036555859544705449, 9.12998324169211696274321649626, 9.89897453820136191856854006861, 10.35952549039076517938382217434, 11.42420667432815509474496673574, 12.28399696555931601016074053374, 13.46456708296425986693706851555, 13.742769231173654000342082869048, 14.723253044737854982730941638439, 15.61429943211640481373484297782, 16.689283012311969256519469491695, 17.333655523039817545639576980713, 18.44683393763953709393951697018, 19.54206741326697045565212609371, 19.85108488996714739199373875321, 20.28760467474924175171577044276, 21.49347863497248352493748031795, 22.12821643644870229332912269762

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