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Properties

Label 1-669-669.374-r0-0-0
Degree $1$
Conductor $669$
Sign $0.999 + 0.00818i$
Analytic cond. $3.10682$
Root an. cond. $3.10682$
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  + (0.967 − 0.251i)2-s + (0.873 − 0.487i)4-s + (−0.238 + 0.971i)5-s + (−0.292 − 0.956i)7-s + (0.721 − 0.691i)8-s + (0.0141 + 0.999i)10-s + (0.0988 + 0.995i)11-s + (−0.778 + 0.628i)13-s + (−0.524 − 0.851i)14-s + (0.524 − 0.851i)16-s + (0.911 + 0.411i)17-s + (0.899 − 0.437i)19-s + (0.265 + 0.964i)20-s + (0.346 + 0.938i)22-s + (0.960 − 0.279i)23-s + ⋯
L(s)  = 1  + (0.967 − 0.251i)2-s + (0.873 − 0.487i)4-s + (−0.238 + 0.971i)5-s + (−0.292 − 0.956i)7-s + (0.721 − 0.691i)8-s + (0.0141 + 0.999i)10-s + (0.0988 + 0.995i)11-s + (−0.778 + 0.628i)13-s + (−0.524 − 0.851i)14-s + (0.524 − 0.851i)16-s + (0.911 + 0.411i)17-s + (0.899 − 0.437i)19-s + (0.265 + 0.964i)20-s + (0.346 + 0.938i)22-s + (0.960 − 0.279i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(669\)    =    \(3 \cdot 223\)
Sign: $0.999 + 0.00818i$
Analytic conductor: \(3.10682\)
Root analytic conductor: \(3.10682\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{669} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 669,\ (0:\ ),\ 0.999 + 0.00818i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.583392728 + 0.01057154078i\)
\(L(\frac12)\) \(\approx\) \(2.583392728 + 0.01057154078i\)
\(L(1)\) \(\approx\) \(1.850983287 - 0.08089335848i\)
\(L(1)\) \(\approx\) \(1.850983287 - 0.08089335848i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
223 \( 1 \)
good2 \( 1 + (0.967 - 0.251i)T \)
5 \( 1 + (-0.238 + 0.971i)T \)
7 \( 1 + (-0.292 - 0.956i)T \)
11 \( 1 + (0.0988 + 0.995i)T \)
13 \( 1 + (-0.778 + 0.628i)T \)
17 \( 1 + (0.911 + 0.411i)T \)
19 \( 1 + (0.899 - 0.437i)T \)
23 \( 1 + (0.960 - 0.279i)T \)
29 \( 1 + (0.999 + 0.0282i)T \)
31 \( 1 + (0.319 + 0.947i)T \)
37 \( 1 + (0.812 + 0.583i)T \)
41 \( 1 + (-0.942 - 0.333i)T \)
43 \( 1 + (0.844 + 0.536i)T \)
47 \( 1 + (-0.899 - 0.437i)T \)
53 \( 1 + (0.398 - 0.916i)T \)
59 \( 1 + (0.985 - 0.169i)T \)
61 \( 1 + (-0.155 - 0.987i)T \)
67 \( 1 + (0.0141 - 0.999i)T \)
71 \( 1 + (-0.346 + 0.938i)T \)
73 \( 1 + (-0.990 + 0.141i)T \)
79 \( 1 + (0.182 + 0.983i)T \)
83 \( 1 + (-0.844 + 0.536i)T \)
89 \( 1 + (0.238 + 0.971i)T \)
97 \( 1 + (-0.960 - 0.279i)T \)
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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

−22.76100100067957113218173945091, −21.9933534782405716369100455134, −21.23546866534135340375791128888, −20.59037039091737473768896024220, −19.62334833635288591729645419473, −18.930430393475076443378786105914, −17.63323703255308589498829185447, −16.62025087913589023233379699912, −16.15822709330473748369854495033, −15.33357971763879777651841419290, −14.51672939384783867307586763476, −13.50260651138286196237401009654, −12.79662573384405202019336943051, −11.961862550207506617187019813029, −11.55857532384605728931561389666, −10.10554282619171410236825687761, −9.04214018189534643307681552285, −8.124889055534635366647689430396, −7.35825413109684241423194270400, −5.907728702630237238891531453234, −5.49767148291546764015688496571, −4.61299997149718411498991777796, −3.34943940807954876545929647652, −2.66425033750637879943430205415, −1.1032115023167052722028613819, 1.2913873234889591659455588432, 2.607341328104463412474612258680, 3.38344820513870746260589931397, 4.35089540230372689337969969661, 5.170026399234790229993156207673, 6.72774743063317401909276317929, 6.886940402134167028042627483760, 7.84611769587834053467413522886, 9.77301827713272833305256991405, 10.14702686774005033056486256927, 11.13985122167611363535019418868, 11.94089403715735014783384746841, 12.74378150815692363217771749822, 13.78825467001536987946904929328, 14.42260151891815908682453605414, 15.032321816551117098064253042509, 16.000498880959271201617356925933, 16.88820170557916150610552289505, 17.8568575226003812631453530111, 19.06549644839046048482001978940, 19.59322826032559579934811728523, 20.345606531527054630747970281530, 21.3038336980702001795317328193, 22.07215866398439985201225360276, 22.91644636030845009844734581908

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