↑

Properties

Label 1-775-775.102-r1-0-0
Degree $1$
Conductor $775$
Sign $-0.226 + 0.974i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.866 + 0.5i)3-s + (−0.309 + 0.951i)4-s + (0.913 + 0.406i)6-s + (0.743 + 0.669i)7-s + (0.951 − 0.309i)8-s + (0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (−0.207 − 0.978i)12-s + (−0.406 − 0.913i)13-s + (0.104 − 0.994i)14-s + (−0.809 − 0.587i)16-s + (0.743 + 0.669i)17-s + (−0.994 + 0.104i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.866 + 0.5i)3-s + (−0.309 + 0.951i)4-s + (0.913 + 0.406i)6-s + (0.743 + 0.669i)7-s + (0.951 − 0.309i)8-s + (0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (−0.207 − 0.978i)12-s + (−0.406 − 0.913i)13-s + (0.104 − 0.994i)14-s + (−0.809 − 0.587i)16-s + (0.743 + 0.669i)17-s + (−0.994 + 0.104i)18-s + (0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.226 + 0.974i$
Analytic conductor: \(83.2853\)
Root analytic conductor: \(83.2853\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 775,\ (1:\ ),\ -0.226 + 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5332885389 + 0.6713096700i\)
\(L(\frac12)\) \(\approx\) \(0.5332885389 + 0.6713096700i\)
\(L(1)\) \(\approx\) \(0.6385759610 + 0.05506796904i\)
\(L(1)\) \(\approx\) \(0.6385759610 + 0.05506796904i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−22.147215932134785714445581244495, −21.21571468922306172976000888600, −20.09377221727507644088653056417, −19.18701310939837168860658726318, −18.42044971798018995656103603881, −17.90952619498606830437307878046, −17.00489557564421864506581960871, −16.41659661712223163593939535785, −15.85545725686060008483834382808, −14.40097105613323384069792420999, −14.006159507851654008404728397635, −13.06363112897787795998009098354, −11.80446796328401332785070543930, −11.04756453285401430328050764042, −10.41746838019193678177024165097, −9.33311066390605938720772754490, −8.27149042978191482597930242489, −7.40283769777064436204846735186, −6.94063588751504219086467293842, −5.78410019919779549697534378056, −5.11300206167798300217760328589, −4.23641043019381253944379798911, −2.30359257529811714784493889515, −1.07797929892145979062554703033, −0.35848978252362850663727272566, 1.03367747018688721641074508909, 2.05520038829957551637608130437, 3.26273778106513463556151380779, 4.341769818405400437339405224698, 5.19571748778337429988916877752, 6.054853163178589937680046519424, 7.71690775271689151102326256008, 7.98207056569679648831344544403, 9.56693162492260485459958070337, 9.85910703539131155760071180808, 10.83638721457551801955960645668, 11.55828674487640346662264131398, 12.39066666247517894309025267228, 12.78930991004044899627644739575, 14.31506432046833318033439756422, 15.28012739957216410255009146853, 15.98199389177089838269266104221, 17.11337654257403044043868534598, 17.59239321803631978916420001756, 18.243856128693481385960395948191, 18.99725988287261063481643650053, 20.209542004267156675019172529052, 20.827676210162250695690894344901, 21.45500973277077052254778188317, 22.244764809990937931422308380711

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