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Properties

Label 1-4033-4033.3211-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.303 - 0.952i$
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  + (0.766 − 0.642i)2-s + (0.286 − 0.957i)3-s + (0.173 − 0.984i)4-s + (−0.998 + 0.0581i)5-s + (−0.396 − 0.918i)6-s + (0.893 + 0.448i)7-s + (−0.5 − 0.866i)8-s + (−0.835 − 0.549i)9-s + (−0.727 + 0.686i)10-s + (0.727 + 0.686i)11-s + (−0.893 − 0.448i)12-s + (0.893 + 0.448i)13-s + (0.973 − 0.230i)14-s + (−0.230 + 0.973i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.286 − 0.957i)3-s + (0.173 − 0.984i)4-s + (−0.998 + 0.0581i)5-s + (−0.396 − 0.918i)6-s + (0.893 + 0.448i)7-s + (−0.5 − 0.866i)8-s + (−0.835 − 0.549i)9-s + (−0.727 + 0.686i)10-s + (0.727 + 0.686i)11-s + (−0.893 − 0.448i)12-s + (0.893 + 0.448i)13-s + (0.973 − 0.230i)14-s + (−0.230 + 0.973i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.303 - 0.952i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.303 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.771448319 - 2.422540301i\)
\(L(\frac12)\) \(\approx\) \(1.771448319 - 2.422540301i\)
\(L(1)\) \(\approx\) \(1.365363626 - 1.052287214i\)
\(L(1)\) \(\approx\) \(1.365363626 - 1.052287214i\)

Euler product

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

−18.72882615173112784267819207876, −17.65949745029317541021269942942, −17.03993762645010324742802671232, −16.35977081578194897471223272078, −15.94447090982469538319451389880, −15.233292047747304868603287436153, −14.62081155666785868203877093165, −14.124655183089756221902427792059, −13.55008425216612295716409881089, −12.483594260204499543682290873039, −11.744070551950881212738694811508, −11.17724688897558941664361288915, −10.70417786181200437542804486249, −9.56452502723717167951682712256, −8.55745414508960086992811548751, −8.1841121148163207861228881328, −7.720062938767764866057130312540, −6.67180838392286590771504890953, −5.7484691036600389685267596917, −5.192282563555286226739173943252, −4.1675898682541952060855481730, −3.917851624719265451532531104501, −3.35498502956055299982226907727, −2.286706148499102304145840151032, −0.86385575812596473832858011798, 0.83310567342039789768637820464, 1.57574249069575449908383441743, 2.23286236255869820253950907597, 3.151676675920584587995965094562, 3.978615350518274690006334411039, 4.49356579415743049762634559661, 5.50011650149813974001518171273, 6.33475889572045407825364434628, 6.91901295139880360077175099784, 7.73746011509774368854312154255, 8.653306079122493732156510761884, 8.91953188804721192003034438104, 10.29066346471109961348042639135, 10.9571509607162283591047746508, 11.70178011817338083511297354565, 12.204406499753209394971154553178, 12.45993766286384163126036973827, 13.480618730366217333794981542852, 14.22895878696017551115784934508, 14.69427586389102657566575834667, 15.15000429537343429036113305878, 16.04940141348021921533888578253, 16.94826978252385360245841659046, 17.99183590932132888907876048746, 18.39350891604425006729608285660

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