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Properties

Label 1-801-801.250-r0-0-0
Degree $1$
Conductor $801$
Sign $0.920 + 0.391i$
Analytic cond. $3.71982$
Root an. cond. $3.71982$
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  + (0.0475 − 0.998i)2-s + (−0.995 − 0.0950i)4-s + (0.786 + 0.618i)5-s + (−0.371 − 0.928i)7-s + (−0.142 + 0.989i)8-s + (0.654 − 0.755i)10-s + (−0.786 + 0.618i)11-s + (−0.814 + 0.580i)13-s + (−0.945 + 0.327i)14-s + (0.981 + 0.189i)16-s + (−0.841 + 0.540i)17-s + (0.755 − 0.654i)19-s + (−0.723 − 0.690i)20-s + (0.580 + 0.814i)22-s + (−0.189 − 0.981i)23-s + ⋯
L(s)  = 1  + (0.0475 − 0.998i)2-s + (−0.995 − 0.0950i)4-s + (0.786 + 0.618i)5-s + (−0.371 − 0.928i)7-s + (−0.142 + 0.989i)8-s + (0.654 − 0.755i)10-s + (−0.786 + 0.618i)11-s + (−0.814 + 0.580i)13-s + (−0.945 + 0.327i)14-s + (0.981 + 0.189i)16-s + (−0.841 + 0.540i)17-s + (0.755 − 0.654i)19-s + (−0.723 − 0.690i)20-s + (0.580 + 0.814i)22-s + (−0.189 − 0.981i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 801 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 801 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(801\)    =    \(3^{2} \cdot 89\)
Sign: $0.920 + 0.391i$
Analytic conductor: \(3.71982\)
Root analytic conductor: \(3.71982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{801} (250, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 801,\ (0:\ ),\ 0.920 + 0.391i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9488124202 + 0.1932378965i\)
\(L(\frac12)\) \(\approx\) \(0.9488124202 + 0.1932378965i\)
\(L(1)\) \(\approx\) \(0.8830935410 - 0.2394308354i\)
\(L(1)\) \(\approx\) \(0.8830935410 - 0.2394308354i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.0475 - 0.998i)T \)
5 \( 1 + (0.786 + 0.618i)T \)
7 \( 1 + (-0.371 - 0.928i)T \)
11 \( 1 + (-0.786 + 0.618i)T \)
13 \( 1 + (-0.814 + 0.580i)T \)
17 \( 1 + (-0.841 + 0.540i)T \)
19 \( 1 + (0.755 - 0.654i)T \)
23 \( 1 + (-0.189 - 0.981i)T \)
29 \( 1 + (0.371 + 0.928i)T \)
31 \( 1 + (0.945 - 0.327i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.0950 + 0.995i)T \)
43 \( 1 + (0.618 + 0.786i)T \)
47 \( 1 + (-0.580 + 0.814i)T \)
53 \( 1 + (-0.415 + 0.909i)T \)
59 \( 1 + (0.0950 - 0.995i)T \)
61 \( 1 + (0.690 - 0.723i)T \)
67 \( 1 + (0.580 + 0.814i)T \)
71 \( 1 + (0.142 + 0.989i)T \)
73 \( 1 + (-0.654 + 0.755i)T \)
79 \( 1 + (-0.981 - 0.189i)T \)
83 \( 1 + (-0.458 + 0.888i)T \)
97 \( 1 + (0.928 - 0.371i)T \)
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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

−22.3052951787337014779776979572, −21.55379331006328551008377843739, −20.899927174425131842907203874548, −19.64586204479606587204963989556, −18.8013899180347885487676326292, −17.84966288781241487567233855661, −17.53220939056269979001323268681, −16.350832106999178728157006642959, −15.84288174787525828707229633858, −15.140934188261599301767134549934, −13.96528056959724751070872342111, −13.45434824217114750568157811802, −12.621680662466366157451824080507, −11.8471816762877913097839291503, −10.24254786256144836042221388944, −9.60376741624207270414139709583, −8.80394397484028785812205385837, −8.06038644134433315256457545732, −7.0530018434484705884103229657, −5.82065879983462653957225499954, −5.5506534249566483337675402303, −4.66779370685702704564904214272, −3.23617024282986205210870515198, −2.17624840147707895557291096220, −0.45674250885442712911922261605, 1.26062288446455138266742313791, 2.4284176346875552367545111794, 2.9977399144376108308551755798, 4.36617580073011094436426811958, 4.94795720926936752039907379544, 6.33297533090082919368215146096, 7.1260487560901341437719659669, 8.26072163322666292089604604714, 9.54928627341614338473493564337, 9.94124601449767217000548594495, 10.715426520109014472758077762, 11.449146356217085266096033612368, 12.70776553392501626996118335185, 13.17200597531565311362961544652, 14.07638716105899490926030510962, 14.62188239704462641306589253392, 15.83517830130936758348517105037, 17.11132028641631917365757057385, 17.570854938718042297804108405438, 18.388605292910161948278366270566, 19.1823590173459535871999419480, 20.072383826751571777043277143518, 20.582592119274454743706775393562, 21.609812433460587032878903249686, 22.14846034192932620376382082350

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