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Properties

Label 1-4033-4033.1835-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.434 + 0.900i$
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 + (0.835 − 0.549i)3-s + (−0.5 − 0.866i)4-s + (0.802 + 0.597i)5-s + (0.0581 + 0.998i)6-s + (0.396 + 0.918i)7-s + 8-s + (0.396 − 0.918i)9-s + (−0.918 + 0.396i)10-s + (0.918 + 0.396i)11-s + (−0.893 − 0.448i)12-s + (0.597 − 0.802i)13-s + (−0.993 − 0.116i)14-s + (0.998 + 0.0581i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.835 − 0.549i)3-s + (−0.5 − 0.866i)4-s + (0.802 + 0.597i)5-s + (0.0581 + 0.998i)6-s + (0.396 + 0.918i)7-s + 8-s + (0.396 − 0.918i)9-s + (−0.918 + 0.396i)10-s + (0.918 + 0.396i)11-s + (−0.893 − 0.448i)12-s + (0.597 − 0.802i)13-s + (−0.993 − 0.116i)14-s + (0.998 + 0.0581i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.434 + 0.900i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1835, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.434 + 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.142446906 + 1.345858384i\)
\(L(\frac12)\) \(\approx\) \(2.142446906 + 1.345858384i\)
\(L(1)\) \(\approx\) \(1.320283420 + 0.4941038112i\)
\(L(1)\) \(\approx\) \(1.320283420 + 0.4941038112i\)

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 \( 1 + (0.835 - 0.549i)T \)
5 \( 1 + (0.802 + 0.597i)T \)
7 \( 1 + (0.396 + 0.918i)T \)
11 \( 1 + (0.918 + 0.396i)T \)
13 \( 1 + (0.597 - 0.802i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (0.116 - 0.993i)T \)
31 \( 1 + (-0.727 - 0.686i)T \)
41 \( 1 + (-0.342 + 0.939i)T \)
43 \( 1 + (0.984 - 0.173i)T \)
47 \( 1 + (-0.998 - 0.0581i)T \)
53 \( 1 + (0.998 - 0.0581i)T \)
59 \( 1 + (-0.0581 + 0.998i)T \)
61 \( 1 + (0.727 - 0.686i)T \)
67 \( 1 + (0.448 + 0.893i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.993 - 0.116i)T \)
79 \( 1 + (-0.835 - 0.549i)T \)
83 \( 1 + (0.686 + 0.727i)T \)
89 \( 1 + (0.918 - 0.396i)T \)
97 \( 1 + (0.727 - 0.686i)T \)
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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

−18.41891568253580904662029754763, −17.68824382752108896518849470359, −16.98998448617893510958623219625, −16.41899331984588489094222648869, −16.01974805837003179275701632317, −14.555340323839640324078196085168, −14.03479719864689507742671948691, −13.71250434075485031836592316122, −12.97120977403339762360671558474, −12.12912471681963655395521575030, −11.22943688230103178822917454059, −10.69296013657161455809700974705, −9.95678436262015469035142862226, −9.304022977433401328708616190163, −8.86009471932291055920492871557, −8.26108974017828031016112196291, −7.356377144146795389049644997, −6.589164743495327504237822833947, −5.25768274858517226824336727435, −4.58640358776631987957458366239, −3.840373232407814180927717040732, −3.37958362592249954040087591913, −2.1467175850661216289571073400, −1.66513858471591639042065851365, −0.86010894217419703265386566999, 1.03487622870422560100475703349, 1.91472625783446960113303340221, 2.30976160160966522762901879441, 3.56265901653742747710319519924, 4.33260955796118823408532931037, 5.61607421558824056174609675754, 6.11161252327368324160274205818, 6.55044208839962399177139660349, 7.504153698359994186303617849292, 8.27686662800782698661907208372, 8.59300053211569476992588091890, 9.50414154286788521638774828937, 9.97151856506259071507301917144, 10.773230266470480999867411869662, 11.77018620548189403161949656844, 12.75552594318002542130944570475, 13.20998885151358263277750491403, 14.17180036368647411339365558268, 14.6507084516430520920419398902, 14.96795503611173883128881170496, 15.63340750239247965081039286131, 16.72612831037376987925955225727, 17.429685476799601328878567890323, 17.9273549614075635079192027594, 18.52972937526553352082347568416

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