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Properties

Label 1-42e2-1764.131-r1-0-0
Degree $1$
Conductor $1764$
Sign $-0.645 - 0.763i$
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  + (−0.988 + 0.149i)5-s + (−0.733 + 0.680i)11-s + (0.733 − 0.680i)13-s + (0.826 − 0.563i)17-s + (−0.5 − 0.866i)19-s + (0.0747 − 0.997i)23-s + (0.955 − 0.294i)25-s + (−0.826 + 0.563i)29-s + 31-s + (0.0747 + 0.997i)37-s + (0.365 − 0.930i)41-s + (−0.365 − 0.930i)43-s + (0.222 + 0.974i)47-s + (−0.0747 + 0.997i)53-s + (0.623 − 0.781i)55-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)5-s + (−0.733 + 0.680i)11-s + (0.733 − 0.680i)13-s + (0.826 − 0.563i)17-s + (−0.5 − 0.866i)19-s + (0.0747 − 0.997i)23-s + (0.955 − 0.294i)25-s + (−0.826 + 0.563i)29-s + 31-s + (0.0747 + 0.997i)37-s + (0.365 − 0.930i)41-s + (−0.365 − 0.930i)43-s + (0.222 + 0.974i)47-s + (−0.0747 + 0.997i)53-s + (0.623 − 0.781i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.645 - 0.763i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ -0.645 - 0.763i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3179815088 - 0.6850964166i\)
\(L(\frac12)\) \(\approx\) \(0.3179815088 - 0.6850964166i\)
\(L(1)\) \(\approx\) \(0.8280872888 - 0.06177852610i\)
\(L(1)\) \(\approx\) \(0.8280872888 - 0.06177852610i\)

Euler product

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

−20.35764186803990638390623859411, −19.3314602142748704167366299958, −19.01997941672616796364961897168, −18.33293692882145543226069679787, −17.28840794614245422246497175935, −16.35869815528134768203210016678, −16.11254787712199002606042461276, −15.139244443606301213179735285124, −14.53066561469557518794644422030, −13.517272631948592845433643111966, −12.90933948511501290764591557425, −11.979966597492702430618087725300, −11.37072841170809747388639362635, −10.68463908584146205733898897948, −9.77339084241669496972858014898, −8.77837249069234190051899895242, −8.03600371007209732310303745603, −7.61755152396164666410906030964, −6.403220208133306506989399764191, −5.71697872401137627694644046447, −4.70968721059915334399352519694, −3.747557497031431900570488594705, −3.30269516674297649872543041807, −1.94784105554916908622769483226, −0.916587818030450982216110925, 0.17934461876811695245324083008, 1.09022778791705902126559432032, 2.54376471552227013171396860739, 3.164619503454279301863995688831, 4.22319565750004013291429679999, 4.89668312161081658011709049429, 5.85661530649219541179417182362, 6.93579377363158134703812723214, 7.53528723395795235747361451249, 8.309113309656283265204869417700, 9.01088073475804232923405294764, 10.22065494760787511226323566071, 10.68652865493633046063194208334, 11.55216771877405885970700188059, 12.34750126451998435676389335084, 12.940823915112917207025112362191, 13.83002652338014584810953030931, 14.81086216082693906711966283899, 15.40333146712375735236740255186, 15.92481641728443525158460348557, 16.78084487149628360830286029037, 17.651780368666854591670172691383, 18.52912982433771333754796917918, 18.87650049683118362549921138216, 19.88200710169515406650504045126

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