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Properties

Label 1-516-516.239-r0-0-0
Degree $1$
Conductor $516$
Sign $-0.628 - 0.778i$
Analytic cond. $2.39629$
Root an. cond. $2.39629$
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 + (0.5 − 0.866i)7-s + (−0.900 − 0.433i)11-s + (−0.988 + 0.149i)13-s + (−0.365 − 0.930i)17-s + (−0.826 − 0.563i)19-s + (0.0747 + 0.997i)23-s + (−0.733 − 0.680i)25-s + (−0.955 − 0.294i)29-s + (0.733 − 0.680i)31-s + (0.623 + 0.781i)35-s + (−0.5 − 0.866i)37-s + (0.222 + 0.974i)41-s + (−0.900 + 0.433i)47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (−0.365 + 0.930i)5-s + (0.5 − 0.866i)7-s + (−0.900 − 0.433i)11-s + (−0.988 + 0.149i)13-s + (−0.365 − 0.930i)17-s + (−0.826 − 0.563i)19-s + (0.0747 + 0.997i)23-s + (−0.733 − 0.680i)25-s + (−0.955 − 0.294i)29-s + (0.733 − 0.680i)31-s + (0.623 + 0.781i)35-s + (−0.5 − 0.866i)37-s + (0.222 + 0.974i)41-s + (−0.900 + 0.433i)47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 516 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 516 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(516\)    =    \(2^{2} \cdot 3 \cdot 43\)
Sign: $-0.628 - 0.778i$
Analytic conductor: \(2.39629\)
Root analytic conductor: \(2.39629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{516} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 516,\ (0:\ ),\ -0.628 - 0.778i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1993717783 - 0.4170864063i\)
\(L(\frac12)\) \(\approx\) \(0.1993717783 - 0.4170864063i\)
\(L(1)\) \(\approx\) \(0.7485182766 - 0.08010232113i\)
\(L(1)\) \(\approx\) \(0.7485182766 - 0.08010232113i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
43 \( 1 \)
good5 \( 1 + (-0.365 + 0.930i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.900 - 0.433i)T \)
13 \( 1 + (-0.988 + 0.149i)T \)
17 \( 1 + (-0.365 - 0.930i)T \)
19 \( 1 + (-0.826 - 0.563i)T \)
23 \( 1 + (0.0747 + 0.997i)T \)
29 \( 1 + (-0.955 - 0.294i)T \)
31 \( 1 + (0.733 - 0.680i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (-0.900 + 0.433i)T \)
53 \( 1 + (0.988 + 0.149i)T \)
59 \( 1 + (0.623 - 0.781i)T \)
61 \( 1 + (-0.733 - 0.680i)T \)
67 \( 1 + (-0.826 - 0.563i)T \)
71 \( 1 + (0.0747 - 0.997i)T \)
73 \( 1 + (-0.988 + 0.149i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.955 - 0.294i)T \)
89 \( 1 + (-0.955 + 0.294i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−24.11141425196242505037982446390, −23.09764382386010361171724825828, −22.1306414065540974877070721987, −21.158590230865146830799325562593, −20.6766583199313966050298823029, −19.6259372950021362906609264342, −18.886097828200764779288598470874, −17.86678235895600734196754952269, −17.08590102964494878441468113673, −16.21924036274742482623976602899, −15.15943894850980344016747524002, −14.8159119879254041427481515381, −13.32441670212814946446403077418, −12.46736237795673893473584230933, −12.0830176940972279977616730434, −10.79736920204948731411835742671, −9.90382666762618973557849217761, −8.63342947979462893906027373446, −8.28528009945114443942262059816, −7.143134904112022586269216525963, −5.756502371666710440502956875583, −4.98011696751539767349578838943, −4.17141101792496785974775127779, −2.60403934983169745180231348562, −1.670791161356940949976919787925, 0.23008442224580908453333993719, 2.11890460786574593904085030398, 3.07304382849209290085259386692, 4.228412947983997219693755674106, 5.15662496860983864321551670079, 6.48886476229162813512135595657, 7.444642600936010205663880819765, 7.88770378764687865296531792924, 9.34910164918478440264736803194, 10.32796701101659069436196112298, 11.095630517359704980477303852950, 11.71387112144642234339970824429, 13.126672691704534303421813587598, 13.81447609493649144629838553401, 14.7460137752745690570188522743, 15.43901310414931901963978450198, 16.46039801862713460074145147830, 17.45562763615674743106083325079, 18.12117003174150863916799452165, 19.14595088974889948602702971491, 19.733459808425852913361432527307, 20.815804012352666249969474704861, 21.56733813691066449700091641005, 22.51306077810880228729254334605, 23.28499155163564909284294834242

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