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Properties

Label 1-669-669.182-r0-0-0
Degree $1$
Conductor $669$
Sign $-0.546 - 0.837i$
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  + (0.828 − 0.559i)2-s + (0.372 − 0.927i)4-s + (0.210 − 0.977i)5-s + (0.985 − 0.169i)7-s + (−0.210 − 0.977i)8-s + (−0.372 − 0.927i)10-s + (−0.450 + 0.892i)11-s + (−0.873 − 0.487i)13-s + (0.721 − 0.691i)14-s + (−0.721 − 0.691i)16-s + (0.450 − 0.892i)17-s + (0.942 + 0.333i)19-s + (−0.828 − 0.559i)20-s + (0.127 + 0.991i)22-s + (0.210 − 0.977i)23-s + ⋯
L(s)  = 1  + (0.828 − 0.559i)2-s + (0.372 − 0.927i)4-s + (0.210 − 0.977i)5-s + (0.985 − 0.169i)7-s + (−0.210 − 0.977i)8-s + (−0.372 − 0.927i)10-s + (−0.450 + 0.892i)11-s + (−0.873 − 0.487i)13-s + (0.721 − 0.691i)14-s + (−0.721 − 0.691i)16-s + (0.450 − 0.892i)17-s + (0.942 + 0.333i)19-s + (−0.828 − 0.559i)20-s + (0.127 + 0.991i)22-s + (0.210 − 0.977i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(669\)    =    \(3 \cdot 223\)
Sign: $-0.546 - 0.837i$
Analytic conductor: \(3.10682\)
Root analytic conductor: \(3.10682\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{669} (182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 669,\ (0:\ ),\ -0.546 - 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.173925417 - 2.168446355i\)
\(L(\frac12)\) \(\approx\) \(1.173925417 - 2.168446355i\)
\(L(1)\) \(\approx\) \(1.434656307 - 1.068640118i\)
\(L(1)\) \(\approx\) \(1.434656307 - 1.068640118i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
223 \( 1 \)
good2 \( 1 + (0.828 - 0.559i)T \)
5 \( 1 + (0.210 - 0.977i)T \)
7 \( 1 + (0.985 - 0.169i)T \)
11 \( 1 + (-0.450 + 0.892i)T \)
13 \( 1 + (-0.873 - 0.487i)T \)
17 \( 1 + (0.450 - 0.892i)T \)
19 \( 1 + (0.942 + 0.333i)T \)
23 \( 1 + (0.210 - 0.977i)T \)
29 \( 1 + (0.721 + 0.691i)T \)
31 \( 1 + (-0.594 + 0.803i)T \)
37 \( 1 + (-0.450 - 0.892i)T \)
41 \( 1 + (0.967 - 0.251i)T \)
43 \( 1 + (-0.911 + 0.411i)T \)
47 \( 1 + (-0.942 + 0.333i)T \)
53 \( 1 + (0.996 + 0.0848i)T \)
59 \( 1 + (-0.127 + 0.991i)T \)
61 \( 1 + (-0.873 - 0.487i)T \)
67 \( 1 + (-0.372 + 0.927i)T \)
71 \( 1 + (-0.127 + 0.991i)T \)
73 \( 1 + (0.778 - 0.628i)T \)
79 \( 1 + (0.967 - 0.251i)T \)
83 \( 1 + (0.911 + 0.411i)T \)
89 \( 1 + (-0.210 - 0.977i)T \)
97 \( 1 + (-0.210 - 0.977i)T \)
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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

−23.10613515990310743946326429094, −22.1537666143381080693492482662, −21.50087441663828096234740295483, −21.156844081809483856449159252504, −19.851478691694958009412763317965, −18.87359221154651658940056445133, −17.95998683079238722306250356352, −17.265506789871210514096547758102, −16.40116179320840274524007588061, −15.22358059383662769718310987118, −14.885037968036011822545524751246, −13.88866333558806440997547117673, −13.500654444782122062357128016303, −12.088891863814721469703937694337, −11.469809224163345097898946443163, −10.7028847604246866016019018991, −9.47782834661811286792000072416, −8.13574405481494172271719329757, −7.62142544260693728395814963115, −6.61894334202350988031517414938, −5.6739266177572868897353680916, −4.97388245336846114748829222254, −3.7380488450097229739747692990, −2.859879945519363788817562794, −1.859897638454909455196483842728, 0.93222813377850504085725763407, 1.9210805531295366710854036019, 2.936966276143058943107708161363, 4.34818343305828201772653621706, 5.04758307593836960756199009650, 5.44254262628509092470890632906, 7.02855628409293909180163319815, 7.840156853417835655324856520573, 9.09919004147091416908338717167, 9.98398645791844947121603984945, 10.73387268300369496364046650014, 11.99500635725023617295131442712, 12.30670846027935734756376120007, 13.220283858954722995925835836048, 14.219428082353985436872641415661, 14.71444585308644451091887475407, 15.829359028064374490226385339417, 16.564977621256842099629339749395, 17.797815703840091401368132974586, 18.26307034932531051288868619055, 19.74389696198628895046333588713, 20.19910004936776392857849520256, 20.91605151956506181706239335191, 21.432592022867834819733865278763, 22.58086222918698371742406370050

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