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Properties

Label 1-3332-3332.2515-r1-0-0
Degree $1$
Conductor $3332$
Sign $-0.606 - 0.794i$
Analytic cond. $358.073$
Root an. cond. $358.073$
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)3-s + (−0.365 + 0.930i)5-s + (0.955 − 0.294i)9-s + (0.955 + 0.294i)11-s + (−0.222 − 0.974i)13-s + (0.222 − 0.974i)15-s + (0.5 + 0.866i)19-s + (0.826 + 0.563i)23-s + (−0.733 − 0.680i)25-s + (−0.900 + 0.433i)27-s + (0.900 + 0.433i)29-s + (−0.5 + 0.866i)31-s + (−0.988 − 0.149i)33-s + (−0.0747 − 0.997i)37-s + (0.365 + 0.930i)39-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)3-s + (−0.365 + 0.930i)5-s + (0.955 − 0.294i)9-s + (0.955 + 0.294i)11-s + (−0.222 − 0.974i)13-s + (0.222 − 0.974i)15-s + (0.5 + 0.866i)19-s + (0.826 + 0.563i)23-s + (−0.733 − 0.680i)25-s + (−0.900 + 0.433i)27-s + (0.900 + 0.433i)29-s + (−0.5 + 0.866i)31-s + (−0.988 − 0.149i)33-s + (−0.0747 − 0.997i)37-s + (0.365 + 0.930i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.606 - 0.794i$
Analytic conductor: \(358.073\)
Root analytic conductor: \(358.073\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3332,\ (1:\ ),\ -0.606 - 0.794i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08032239709 - 0.1623307393i\)
\(L(\frac12)\) \(\approx\) \(0.08032239709 - 0.1623307393i\)
\(L(1)\) \(\approx\) \(0.7234862629 + 0.1340412440i\)
\(L(1)\) \(\approx\) \(0.7234862629 + 0.1340412440i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-0.988 + 0.149i)T \)
5 \( 1 + (-0.365 + 0.930i)T \)
11 \( 1 + (0.955 + 0.294i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.826 + 0.563i)T \)
29 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.0747 - 0.997i)T \)
41 \( 1 + (-0.623 - 0.781i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (0.733 - 0.680i)T \)
53 \( 1 + (0.0747 - 0.997i)T \)
59 \( 1 + (-0.365 - 0.930i)T \)
61 \( 1 + (-0.0747 - 0.997i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (0.733 + 0.680i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (0.955 - 0.294i)T \)
97 \( 1 - T \)
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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

−18.97896080994357161290580058085, −18.185996599409567863962026239576, −17.27702324061329179236788806030, −16.813460533467826836600731779476, −16.46615663102251740327305032403, −15.58072737353973752052040547868, −14.96572959272947470576104751313, −13.78890509426176413412192977400, −13.35560407106493254106825697162, −12.43063201449369553034653797610, −11.82180382953222815306943076958, −11.51100085742347012307114031788, −10.64992777894303373288956054332, −9.629963488289081057576249857879, −9.10346850803776464326476884538, −8.33082861974157053755143295761, −7.31491990607274101552524816509, −6.73462679447983065900192156818, −5.99138362469544076451600312642, −5.09320518495017305550464031515, −4.49508455244359833093218826141, −3.93713738280920594692179432158, −2.64657261866545880777487614278, −1.412394645629559453113991925132, −0.95076976485729832639100384057, 0.04139430702953612520130087393, 1.04375324286663085634876865058, 1.96660788820568484371458966335, 3.32141171988612173022997288124, 3.64009560585421812309023120798, 4.76224010254993544466931348785, 5.44417824314366481478453492800, 6.25316889189306646847330720805, 6.967064370616167826531456412648, 7.43120897125933621213691515401, 8.41853293690340503182687688326, 9.50010994392325725910822452022, 10.124140195151649813882676327202, 10.76647667103206042825640375331, 11.34711116467450584543529337451, 12.16486472578045088965957425347, 12.501195879347558360177253637979, 13.58467140387508922565667630049, 14.48427224878836894024001963027, 14.95467253054150846688896805182, 15.770830729418259993860619139241, 16.27166025372618002800080955492, 17.29696474947075335577817682877, 17.60582501807002237504639457805, 18.36424600706007008021733781598

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