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Properties

Label 1-684-684.427-r1-0-0
Degree $1$
Conductor $684$
Sign $-0.486 - 0.873i$
Analytic cond. $73.5060$
Root an. cond. $73.5060$
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  + (0.173 − 0.984i)5-s − 7-s + (0.5 + 0.866i)11-s + (0.766 − 0.642i)13-s + (−0.939 + 0.342i)17-s + (0.939 + 0.342i)23-s + (−0.939 − 0.342i)25-s + (0.766 − 0.642i)29-s + (0.5 − 0.866i)31-s + (−0.173 + 0.984i)35-s + 37-s + (−0.939 + 0.342i)41-s + (0.939 − 0.342i)43-s + (−0.766 + 0.642i)47-s + 49-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)5-s − 7-s + (0.5 + 0.866i)11-s + (0.766 − 0.642i)13-s + (−0.939 + 0.342i)17-s + (0.939 + 0.342i)23-s + (−0.939 − 0.342i)25-s + (0.766 − 0.642i)29-s + (0.5 − 0.866i)31-s + (−0.173 + 0.984i)35-s + 37-s + (−0.939 + 0.342i)41-s + (0.939 − 0.342i)43-s + (−0.766 + 0.642i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.486 - 0.873i$
Analytic conductor: \(73.5060\)
Root analytic conductor: \(73.5060\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 684,\ (1:\ ),\ -0.486 - 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7034282935 - 1.196437108i\)
\(L(\frac12)\) \(\approx\) \(0.7034282935 - 1.196437108i\)
\(L(1)\) \(\approx\) \(0.9583496049 - 0.2552890262i\)
\(L(1)\) \(\approx\) \(0.9583496049 - 0.2552890262i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (0.173 - 0.984i)T \)
7 \( 1 - T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.766 - 0.642i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.766 + 0.642i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (-0.766 - 0.642i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 - T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−22.86309604100121216107562315281, −21.83775964805174324431074231954, −21.52167589412850198111296539694, −20.19617458416369094307177966447, −19.36180043246233619025498042143, −18.77244777554497896518704911203, −18.043804844494490504513379719555, −16.97663219582531732296450549685, −16.13447218429119617655055061288, −15.46927512281936464490249592987, −14.36459525830632505483082085124, −13.719954214802335032686757243752, −12.96790784201355552447820953147, −11.73270492640739911299120381545, −11.02267460106279896057796851384, −10.238939215522861318409962330702, −9.19260721589903474939590625966, −8.53588832556860934874905985245, −6.96407581183669159844777471175, −6.62406786952238480678574786007, −5.75905763103915078207921683202, −4.28888954713876141358038048201, −3.293908491237904950636285355098, −2.61046515667269586906947129922, −1.10313401611887894373351739225, 0.36146502569914372050862320288, 1.45320880844526152219072354764, 2.71422122615399793701334541500, 3.93729357032297959494071495540, 4.72450524934490498263129983952, 5.91963756555871826082760142600, 6.59486813896310017545675496691, 7.79499849613639378527476948058, 8.79951553050255724860808337712, 9.46896778341995692025084890133, 10.267227214465081023635958369811, 11.443129524340003469270701009228, 12.39664110770317164992346339321, 13.11914058334394157685400620896, 13.57772911917310518493212567845, 15.065267286829428630080643667783, 15.62865741523108991848486965695, 16.521491296832426510680461591982, 17.2885847469566428360285307020, 17.9771043740258968889304729946, 19.19970671659619771461559808097, 19.852288101738301559522022575696, 20.52240461047560212733829965845, 21.33585887883471507034548182812, 22.392523392048309319359078758456

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