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Properties

Label 1-42e2-1764.1759-r1-0-0
Degree $1$
Conductor $1764$
Sign $0.00356 - 0.999i$
Analytic cond. $189.568$
Root an. cond. $189.568$
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  + (0.365 + 0.930i)5-s + (−0.955 + 0.294i)11-s + (0.955 − 0.294i)13-s + (0.0747 − 0.997i)17-s + (0.5 + 0.866i)19-s + (−0.826 + 0.563i)23-s + (−0.733 + 0.680i)25-s + (0.0747 − 0.997i)29-s − 31-s + (0.826 + 0.563i)37-s + (−0.988 − 0.149i)41-s + (0.988 − 0.149i)43-s + (0.222 − 0.974i)47-s + (0.826 − 0.563i)53-s + (−0.623 − 0.781i)55-s + ⋯
L(s)  = 1  + (0.365 + 0.930i)5-s + (−0.955 + 0.294i)11-s + (0.955 − 0.294i)13-s + (0.0747 − 0.997i)17-s + (0.5 + 0.866i)19-s + (−0.826 + 0.563i)23-s + (−0.733 + 0.680i)25-s + (0.0747 − 0.997i)29-s − 31-s + (0.826 + 0.563i)37-s + (−0.988 − 0.149i)41-s + (0.988 − 0.149i)43-s + (0.222 − 0.974i)47-s + (0.826 − 0.563i)53-s + (−0.623 − 0.781i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00356 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00356 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.00356 - 0.999i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1759, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ 0.00356 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6758878749 - 0.6734847097i\)
\(L(\frac12)\) \(\approx\) \(0.6758878749 - 0.6734847097i\)
\(L(1)\) \(\approx\) \(0.9854232219 + 0.1054305005i\)
\(L(1)\) \(\approx\) \(0.9854232219 + 0.1054305005i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.365 + 0.930i)T \)
11 \( 1 + (-0.955 + 0.294i)T \)
13 \( 1 + (0.955 - 0.294i)T \)
17 \( 1 + (0.0747 - 0.997i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.826 + 0.563i)T \)
29 \( 1 + (0.0747 - 0.997i)T \)
31 \( 1 - T \)
37 \( 1 + (0.826 + 0.563i)T \)
41 \( 1 + (-0.988 - 0.149i)T \)
43 \( 1 + (0.988 - 0.149i)T \)
47 \( 1 + (0.222 - 0.974i)T \)
53 \( 1 + (0.826 - 0.563i)T \)
59 \( 1 + (-0.623 - 0.781i)T \)
61 \( 1 + (-0.900 + 0.433i)T \)
67 \( 1 - T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (0.955 + 0.294i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.955 - 0.294i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
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

−20.12503309408418801173291677374, −19.80232161638356594649366214467, −18.51508503113774648838282898060, −18.17249661921671728100815924839, −17.260148082148937132475090739235, −16.42152955604688728602171719953, −16.0049395588176305348350599889, −15.18726289923052704378224925046, −14.115615546566243787776930188088, −13.488898126973272355728948272614, −12.807492339836655189620949013020, −12.22471431559097004776240779094, −11.06183088339096894346407511771, −10.58099239228619590045558325457, −9.53143167813330555873673837058, −8.81682574429518499162468819659, −8.213066417075114864932826181374, −7.33983573139991130953337835069, −6.133671744277272959238196748078, −5.650322504233745475317157712922, −4.71019677190792217427941463426, −3.92482268314413568689098145469, −2.83898803315629707169991200531, −1.7985693569369358590768055332, −0.94850680315883510358347264217, 0.18177934081019688587847597512, 1.54940669146357102109868453078, 2.47420821197933421344404689847, 3.268636213087267732165337025124, 4.10563000495448876711326688723, 5.4306036115306318382543244194, 5.83202138751403431261772735709, 6.88075583748030462274304785048, 7.62217129632441585703749174780, 8.28341254128042480423609980003, 9.53637015462743188804719160461, 10.02686118078698683969127169590, 10.819926557295245935781525580493, 11.51358332203159485846015437413, 12.35695423972323492868237416873, 13.53305117714242484282474106963, 13.686694753793645359238983888609, 14.70923512142409093654240355691, 15.47399400625752987175054619991, 16.03193462519828529250279949044, 16.9569566643355170215485162275, 18.03077414160807115478787878571, 18.28541051991002971072357353497, 18.853774754434218470731052727, 20.0107777811705811624144286699

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