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Properties

Label 1-4033-4033.1187-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.789 - 0.613i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
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 + (−0.286 − 0.957i)3-s + (−0.5 − 0.866i)4-s + (−0.893 + 0.448i)5-s + (−0.973 − 0.230i)6-s + (−0.835 − 0.549i)7-s − 8-s + (−0.835 + 0.549i)9-s + (−0.0581 + 0.998i)10-s + (−0.0581 − 0.998i)11-s + (−0.686 + 0.727i)12-s + (−0.893 + 0.448i)13-s + (−0.893 + 0.448i)14-s + (0.686 + 0.727i)15-s + (−0.5 + 0.866i)16-s − 17-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.286 − 0.957i)3-s + (−0.5 − 0.866i)4-s + (−0.893 + 0.448i)5-s + (−0.973 − 0.230i)6-s + (−0.835 − 0.549i)7-s − 8-s + (−0.835 + 0.549i)9-s + (−0.0581 + 0.998i)10-s + (−0.0581 − 0.998i)11-s + (−0.686 + 0.727i)12-s + (−0.893 + 0.448i)13-s + (−0.893 + 0.448i)14-s + (0.686 + 0.727i)15-s + (−0.5 + 0.866i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.789 - 0.613i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.789 - 0.613i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2596145076 - 0.7565928077i\)
\(L(\frac12)\) \(\approx\) \(0.2596145076 - 0.7565928077i\)
\(L(1)\) \(\approx\) \(0.5227920538 - 0.5491195402i\)
\(L(1)\) \(\approx\) \(0.5227920538 - 0.5491195402i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-0.286 - 0.957i)T \)
5 \( 1 + (-0.893 + 0.448i)T \)
7 \( 1 + (-0.835 - 0.549i)T \)
11 \( 1 + (-0.0581 - 0.998i)T \)
13 \( 1 + (-0.893 + 0.448i)T \)
17 \( 1 - T \)
19 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (-0.893 + 0.448i)T \)
31 \( 1 + (0.993 + 0.116i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (0.973 - 0.230i)T \)
53 \( 1 + (-0.286 - 0.957i)T \)
59 \( 1 + (0.686 + 0.727i)T \)
61 \( 1 + (-0.396 + 0.918i)T \)
67 \( 1 + (-0.686 - 0.727i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.893 + 0.448i)T \)
79 \( 1 + (-0.973 + 0.230i)T \)
83 \( 1 + (0.597 + 0.802i)T \)
89 \( 1 + (-0.893 + 0.448i)T \)
97 \( 1 + (0.993 - 0.116i)T \)
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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

−18.65145410790373161192039025661, −17.649459347076660887319551735654, −17.21123039820074194107115016999, −16.51691461516948972306636377986, −15.77957735600928759240109834513, −15.40781671511027976840036073484, −15.084695812504640483214251887818, −14.24435718053472560437689777208, −13.1037146602591088619213568187, −12.68469713995672837023179142379, −11.93947099629482302851314710359, −11.46042028716866121296579764733, −10.35289655581314666361345326261, −9.34123530753672905048250626672, −9.23287215214680198440836874924, −8.26830759589151343015944920683, −7.40875452161196273079194138477, −6.833852951248748079925637026547, −5.90778289620917610885765497384, −5.12975945971158931933238070731, −4.642477440616996709699483440312, −4.03365114260066436272065841411, −3.1115465352796463459859703265, −2.55717006858597008695986615929, −0.482905594319408439173582052400, 0.441135636302694677296015123221, 1.235431925557045877357411739499, 2.43017125587404437992257536820, 3.029877284015256464100592105613, 3.65981729832505399419886259156, 4.57112933663438747021496514265, 5.425554777495964657679578540116, 6.272407856366921412537283385830, 6.934062699787029497048819326449, 7.48070334044632824866875664130, 8.50799019028504367710451822009, 9.23491370458579321747764299666, 10.16410014454425408414054520805, 10.95616404780153615765048208261, 11.378169811017733117294212855049, 12.029026947367532561880855305232, 12.63127032701645472540433242651, 13.46657999634845083756952178836, 13.74393609892962639512012132745, 14.59209587082857439127793648752, 15.32641252050823207783325944889, 16.23338345348036241353766446483, 16.83487739380637542144356309186, 17.772949177221726775682937792175, 18.53133382089129339432810999566

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