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Properties

Label 1-663-663.332-r0-0-0
Degree $1$
Conductor $663$
Sign $0.561 - 0.827i$
Analytic cond. $3.07895$
Root an. cond. $3.07895$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.707 + 0.707i)5-s + (−0.965 + 0.258i)7-s − 8-s + (−0.965 − 0.258i)10-s + (0.258 − 0.965i)11-s + (−0.707 − 0.707i)14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)19-s + (−0.258 − 0.965i)20-s + (0.965 − 0.258i)22-s + (−0.965 − 0.258i)23-s − i·25-s + (0.258 − 0.965i)28-s + (0.258 − 0.965i)29-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.707 + 0.707i)5-s + (−0.965 + 0.258i)7-s − 8-s + (−0.965 − 0.258i)10-s + (0.258 − 0.965i)11-s + (−0.707 − 0.707i)14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)19-s + (−0.258 − 0.965i)20-s + (0.965 − 0.258i)22-s + (−0.965 − 0.258i)23-s − i·25-s + (0.258 − 0.965i)28-s + (0.258 − 0.965i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(663\)    =    \(3 \cdot 13 \cdot 17\)
Sign: $0.561 - 0.827i$
Analytic conductor: \(3.07895\)
Root analytic conductor: \(3.07895\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{663} (332, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 663,\ (0:\ ),\ 0.561 - 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2059361702 - 0.1091672773i\)
\(L(\frac12)\) \(\approx\) \(0.2059361702 - 0.1091672773i\)
\(L(1)\) \(\approx\) \(0.6604776004 + 0.4093187867i\)
\(L(1)\) \(\approx\) \(0.6604776004 + 0.4093187867i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + (-0.965 + 0.258i)T \)
11 \( 1 + (0.258 - 0.965i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.965 - 0.258i)T \)
29 \( 1 + (0.258 - 0.965i)T \)
31 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (-0.965 - 0.258i)T \)
41 \( 1 + (0.258 - 0.965i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 - iT \)
53 \( 1 + iT \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.258 - 0.965i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.258 + 0.965i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.258 + 0.965i)T \)
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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

−22.68823596772371874103764834990, −22.32300146917160614812809856184, −21.18849346588781952066157214106, −20.33394580311443296871727291314, −19.7474119221796719318787006057, −19.30400159782975587502614163028, −18.173615275784959605487774887286, −17.23468065396409356569492509236, −16.154345847798370332037501666543, −15.440954184437698994740039475344, −14.56866635174921713399466128357, −13.45263538901476209956976727166, −12.7523814780253426362593987835, −12.21388890117056632743608125406, −11.31543327226775296029004102458, −10.31083263829932585790352656788, −9.46537734911340018818130654277, −8.77431231242731264055686300984, −7.429517862033484896882865271724, −6.43757046362742852273562091769, −5.26802634149766169062192541781, −4.3188486036339573512087177666, −3.70075335882204629642388955660, −2.534190380851908193350581385823, −1.29036191836641756780011951746, 0.100259058088660342619455440732, 2.527612652119486004236767387902, 3.56707210485690733888334385244, 4.02472547406996476301367910953, 5.57688981211094054549487429548, 6.27995007348014048775366184028, 7.010279925009813733338910483653, 8.03945917284797712960136419446, 8.74628414023884730879085485095, 9.89760363306028709433993722318, 10.975409759650942466770836911603, 12.07366197569126813275538810386, 12.60087391448817123471835387955, 13.82763710039864351582314791695, 14.31997054799897200697159613110, 15.36667627454070019421256039162, 15.97509299381709926127106586099, 16.57601204781314995758397789618, 17.626971547036380231766375788820, 18.735222640184800355038217525891, 19.08862031591128234188473914934, 20.21107856916567303197016579542, 21.47562225012958194686164692153, 22.041432057146598142762973851638, 22.8429363832450336510344450851

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