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Properties

Label 1-684-684.367-r1-0-0
Degree $1$
Conductor $684$
Sign $-0.756 + 0.654i$
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 + (0.5 − 0.866i)7-s − 11-s + (0.173 + 0.984i)13-s + (0.173 − 0.984i)17-s + (−0.766 + 0.642i)23-s + (−0.939 − 0.342i)25-s + (0.766 − 0.642i)29-s − 31-s + (−0.766 − 0.642i)35-s + 37-s + (−0.939 + 0.342i)41-s + (−0.766 − 0.642i)43-s + (−0.766 + 0.642i)47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)5-s + (0.5 − 0.866i)7-s − 11-s + (0.173 + 0.984i)13-s + (0.173 − 0.984i)17-s + (−0.766 + 0.642i)23-s + (−0.939 − 0.342i)25-s + (0.766 − 0.642i)29-s − 31-s + (−0.766 − 0.642i)35-s + 37-s + (−0.939 + 0.342i)41-s + (−0.766 − 0.642i)43-s + (−0.766 + 0.642i)47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.756 + 0.654i$
Analytic conductor: \(73.5060\)
Root analytic conductor: \(73.5060\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 684,\ (1:\ ),\ -0.756 + 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08436724990 - 0.2264441110i\)
\(L(\frac12)\) \(\approx\) \(-0.08436724990 - 0.2264441110i\)
\(L(1)\) \(\approx\) \(0.8692784164 - 0.2597058641i\)
\(L(1)\) \(\approx\) \(0.8692784164 - 0.2597058641i\)

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 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 - T \)
13 \( 1 + (0.173 + 0.984i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (0.766 - 0.642i)T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (-0.766 + 0.642i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (-0.766 - 0.642i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−23.064301994767675942562494305294, −21.86504683462088918133425355662, −21.73915958258013212045354523383, −20.61815642749396024614247275550, −19.748022398069962091202758642992, −18.60145044475126072889833267276, −18.25112893210412906486822627398, −17.575971032301469437862077903023, −16.33585452546753340005056366303, −15.32556856292197468945331979490, −14.93076771318176140155919020941, −14.02316834358203636713203427616, −12.95812191258823804217282017426, −12.2388451338850541043004551411, −11.05784364468136023676949285811, −10.54610713257649239964617122624, −9.66098994896381565260905095861, −8.28283902631457136439921642933, −7.910382484197547993360135039990, −6.58528511329917607254256923686, −5.77816109264662873217129070413, −4.96326128718443674462395720749, −3.50574845385659002388673088721, −2.66448063898490433803844463991, −1.745801755347137699865666342637, 0.05475659739119681452856105627, 1.2051783236635054957797489017, 2.20898975740687361580019720507, 3.689880240822133876403586309538, 4.66988953827608410520651533082, 5.27629885619526209962110447564, 6.52170064833970705762240306150, 7.64712589636909005553270958288, 8.238661522190554321648709741421, 9.38947858346068952114557844519, 10.05221628387465549517988002565, 11.20766461615736391169083314236, 11.88817626776622413712631617692, 13.0075989730537091717048458207, 13.659757412832426897155795984966, 14.31895754484476762849484183628, 15.651825620429735084671117232080, 16.30363380848248815263760801816, 16.99487780889064610887491734858, 17.885454736517186252154659794414, 18.631947342404560512828237059437, 19.83483130749117154000483610340, 20.40615484053289378686364075580, 21.12747664638894621903756554618, 21.747891101384380016393506049522

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