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Properties

Label 1-684-684.311-r0-0-0
Degree $1$
Conductor $684$
Sign $0.532 + 0.846i$
Analytic cond. $3.17648$
Root an. cond. $3.17648$
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  − 5-s + (0.5 − 0.866i)7-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)23-s + 25-s − 29-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)35-s + 37-s − 41-s + (0.5 + 0.866i)43-s + 47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  − 5-s + (0.5 − 0.866i)7-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)23-s + 25-s − 29-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)35-s + 37-s − 41-s + (0.5 + 0.866i)43-s + 47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.532 + 0.846i$
Analytic conductor: \(3.17648\)
Root analytic conductor: \(3.17648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 684,\ (0:\ ),\ 0.532 + 0.846i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8409184102 + 0.4642541097i\)
\(L(\frac12)\) \(\approx\) \(0.8409184102 + 0.4642541097i\)
\(L(1)\) \(\approx\) \(0.8711243809 + 0.08570864036i\)
\(L(1)\) \(\approx\) \(0.8711243809 + 0.08570864036i\)

Euler product

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

−22.40340238084872137329399475403, −21.957687732610802108384997106368, −20.84293157990243440283238763560, −20.23269944970304385606239190777, −19.03818077160770162533823430127, −18.79138676161619248853458545294, −17.75011100173345290245125416698, −16.72813659330034848594881732514, −15.93565106418439013651443545458, −15.037657584956292489598320573937, −14.680920223876984711982792009668, −13.299845982639128533980305264941, −12.46019915872104190807484585900, −11.74402648217242112828950282838, −10.92682674455898221640510273659, −10.06200849514213228064082593401, −8.679627302743322084074515266027, −8.18035441070530046010533284970, −7.4338898820333930986611520786, −6.01782921361614689021080952688, −5.322559198550801196533732842135, −4.194839605840139049679553862634, −3.18656122430866331296896117418, −2.20460211001978997794446437296, −0.54254144494500601584163300266, 1.11925479913938727302822299508, 2.44386043297106204182071851144, 3.74057668402912282130308832477, 4.4777935807393651546237541704, 5.292439739540595067208200014086, 6.94232883659637605371788149417, 7.40862894932241022910774981767, 8.15447317658373219376696316644, 9.4100205850869796351674334487, 10.21963646646966047110006858283, 11.284660018456866851125108470679, 11.83975774519875248672361102081, 12.77255790334721910801005271185, 13.84921526798971697077792583003, 14.57929862663208674681199883865, 15.43488255567520227089044448340, 16.282563964316280287867135485767, 17.02413364296642010371371495427, 17.97532941612017975712118001702, 18.79574233337419274330437171639, 19.721668020187775926319102574, 20.329587864882433795686214360090, 21.04288842022304599150864503998, 22.11056740922796947603242351497, 23.12679884764348470285024093569

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