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Properties

Label 1-684-684.491-r0-0-0
Degree $1$
Conductor $684$
Sign $0.654 - 0.756i$
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  + (−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

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

Invariants

Degree: \(1\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.654 - 0.756i$
Analytic conductor: \(3.17648\)
Root analytic conductor: \(3.17648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 684,\ (0:\ ),\ 0.654 - 0.756i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.373953629 - 0.6280562541i\)
\(L(\frac12)\) \(\approx\) \(1.373953629 - 0.6280562541i\)
\(L(1)\) \(\approx\) \(1.138885972 - 0.2176683309i\)
\(L(1)\) \(\approx\) \(1.138885972 - 0.2176683309i\)

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

−22.83527857779255749709507738747, −21.99749522197919943607930782047, −21.35560203423876021551967885810, −20.25921163780826001292551266527, −19.56181753784502573320075270522, −18.76387647260583140791457861391, −17.973925131406941559392375225027, −16.9720482915795367024712028535, −16.51524615220090104110645393448, −15.08984664520891106428362853130, −14.57640279858430327337254479953, −13.9270261979380786989725147972, −12.911040628464685205291276805378, −11.71347283260117011259964045977, −11.06055211381779416271932144578, −10.41744733064503899959243277192, −9.290806998106794063129666537996, −8.36772335680084409342275367492, −7.1175118176544691690694103542, −6.83202933448978637168429299813, −5.64728748945401273698957072472, −4.10479962717191306753221012806, −3.844164643436498561776909934494, −2.34613116151310235080676723421, −1.26748210137464308620672859734, 0.85516071906931977222815412561, 1.96569967013057892515129973681, 3.24088292320316053256887068995, 4.396871824633650653973793014, 5.28031897062116370764781822255, 5.98588961613698210594889614032, 7.373364174926798804491424395006, 8.20393197770991971519679341321, 9.12340597803459294539421551694, 9.58079423719985000160383589690, 11.18877433646118869997798147930, 11.65439976543791477874030984175, 12.63598776291815699629000287583, 13.27189952009547556249765595128, 14.44949980354661219150267214474, 15.218331500996936235911569532182, 15.99701536057467171191734474248, 16.88349630523832596782549828219, 17.65974748592239072214359624480, 18.44707388585958483379740022343, 19.47817230956366058382397281699, 20.21840585663395947587046880165, 20.892366504230438440544632900070, 21.761079443076691318734952853312, 22.57133762411755058956420088318

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