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Properties

Label 1-777-777.143-r1-0-0
Degree $1$
Conductor $777$
Sign $-0.751 - 0.660i$
Analytic cond. $83.5002$
Root an. cond. $83.5002$
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  + (0.984 − 0.173i)2-s + (0.939 − 0.342i)4-s + (−0.984 − 0.173i)5-s + (0.866 − 0.5i)8-s − 10-s + (−0.5 + 0.866i)11-s + (0.984 + 0.173i)13-s + (0.766 − 0.642i)16-s + (−0.642 − 0.766i)17-s + (−0.342 − 0.939i)19-s + (−0.984 + 0.173i)20-s + (−0.342 + 0.939i)22-s + (0.866 − 0.5i)23-s + (0.939 + 0.342i)25-s + 26-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)2-s + (0.939 − 0.342i)4-s + (−0.984 − 0.173i)5-s + (0.866 − 0.5i)8-s − 10-s + (−0.5 + 0.866i)11-s + (0.984 + 0.173i)13-s + (0.766 − 0.642i)16-s + (−0.642 − 0.766i)17-s + (−0.342 − 0.939i)19-s + (−0.984 + 0.173i)20-s + (−0.342 + 0.939i)22-s + (0.866 − 0.5i)23-s + (0.939 + 0.342i)25-s + 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(777\)    =    \(3 \cdot 7 \cdot 37\)
Sign: $-0.751 - 0.660i$
Analytic conductor: \(83.5002\)
Root analytic conductor: \(83.5002\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{777} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 777,\ (1:\ ),\ -0.751 - 0.660i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6934240980 - 1.839397777i\)
\(L(\frac12)\) \(\approx\) \(0.6934240980 - 1.839397777i\)
\(L(1)\) \(\approx\) \(1.434570840 - 0.4199183913i\)
\(L(1)\) \(\approx\) \(1.434570840 - 0.4199183913i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.984 - 0.173i)T \)
5 \( 1 + (-0.984 - 0.173i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.984 + 0.173i)T \)
17 \( 1 + (-0.642 - 0.766i)T \)
19 \( 1 + (-0.342 - 0.939i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.173 - 0.984i)T \)
59 \( 1 + (0.984 - 0.173i)T \)
61 \( 1 + (0.642 - 0.766i)T \)
67 \( 1 + (-0.766 - 0.642i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.984 - 0.173i)T \)
83 \( 1 + (0.173 + 0.984i)T \)
89 \( 1 + (0.642 + 0.766i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−22.62501266575944868453829587280, −21.737096816187262875897632734393, −20.903461430726516357197929483124, −20.301011248351411599256343618291, −19.224842904304112247649154474028, −18.75592713545969448367086495497, −17.42693544493306786625141887379, −16.4030161481596153185798339743, −15.89429874910930189065881183115, −15.083304218895328351809953840648, −14.45209721624845395439938675060, −13.23239113106472361656201725385, −12.92091147326918789934919777767, −11.70711657421094818212960170321, −11.09566684543511069808285585965, −10.50580125161287648356383374171, −8.7789827124462782561656996622, −8.0399504858378697863632865763, −7.25993917790237067865887619664, −6.16091794796582680852693785095, −5.51873542850354664612137347619, −4.18905029361696827641775201461, −3.67362074558763048041642148014, −2.73626351103855143263025464387, −1.34487121372473276271414113157, 0.2967647841563293699103295655, 1.6946516012429455536582980048, 2.84660569495038785590556649895, 3.76420998491513585040166734112, 4.66734206837211182968922704775, 5.27639472616164955460100859979, 6.76801631672775481564768535378, 7.13722524954128874607155814023, 8.324410789463740505277167753081, 9.32355361445926533876904580925, 10.69433265249108915513319174961, 11.18175547722740599780151406898, 12.00783553231170816946966816275, 12.96072423440475582158021387589, 13.3668776998886607751386685345, 14.68479857091240087470052550313, 15.20749579320945102021899990411, 15.97897086311254413194577311784, 16.58487083856337740506241353276, 17.93401327347414591557641419113, 18.797709046020078193458795438527, 19.70751798785533820192458774732, 20.41393535269187732186893528612, 20.87139092727154727315053416077, 22.009735903229141146292101569326

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