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Properties

Label 1-4033-4033.2123-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.973 + 0.227i$
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  + (−0.939 + 0.342i)2-s + (0.993 − 0.116i)3-s + (0.766 − 0.642i)4-s + (−0.957 + 0.286i)5-s + (−0.893 + 0.448i)6-s + (−0.686 + 0.727i)7-s + (−0.5 + 0.866i)8-s + (0.973 − 0.230i)9-s + (0.802 − 0.597i)10-s + (−0.802 − 0.597i)11-s + (0.686 − 0.727i)12-s + (−0.686 + 0.727i)13-s + (0.396 − 0.918i)14-s + (−0.918 + 0.396i)15-s + (0.173 − 0.984i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.993 − 0.116i)3-s + (0.766 − 0.642i)4-s + (−0.957 + 0.286i)5-s + (−0.893 + 0.448i)6-s + (−0.686 + 0.727i)7-s + (−0.5 + 0.866i)8-s + (0.973 − 0.230i)9-s + (0.802 − 0.597i)10-s + (−0.802 − 0.597i)11-s + (0.686 − 0.727i)12-s + (−0.686 + 0.727i)13-s + (0.396 − 0.918i)14-s + (−0.918 + 0.396i)15-s + (0.173 − 0.984i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.973 + 0.227i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.973 + 0.227i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05929960653 + 0.5133527269i\)
\(L(\frac12)\) \(\approx\) \(0.05929960653 + 0.5133527269i\)
\(L(1)\) \(\approx\) \(0.6456679858 + 0.2037124030i\)
\(L(1)\) \(\approx\) \(0.6456679858 + 0.2037124030i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.939 + 0.342i)T \)
3 \( 1 + (0.993 - 0.116i)T \)
5 \( 1 + (-0.957 + 0.286i)T \)
7 \( 1 + (-0.686 + 0.727i)T \)
11 \( 1 + (-0.802 - 0.597i)T \)
13 \( 1 + (-0.686 + 0.727i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.116 + 0.993i)T \)
31 \( 1 + (-0.230 - 0.973i)T \)
41 \( 1 + iT \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (-0.998 + 0.0581i)T \)
53 \( 1 + (0.230 - 0.973i)T \)
59 \( 1 + (0.396 + 0.918i)T \)
61 \( 1 + (-0.549 + 0.835i)T \)
67 \( 1 + (0.957 + 0.286i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (-0.597 + 0.802i)T \)
79 \( 1 + (-0.686 - 0.727i)T \)
83 \( 1 + (0.993 - 0.116i)T \)
89 \( 1 + (0.549 - 0.835i)T \)
97 \( 1 + (-0.957 + 0.286i)T \)
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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

−18.38623387847730589110176681454, −17.5362857412556673332225882586, −16.72694139845865906823103345013, −16.05720719998085315835418787559, −15.6066281273327423555535529197, −15.03275899005522342280439961348, −14.07717493193575244683805096255, −13.17600652078289573854618957574, −12.51748990303728185607097175995, −12.19909123168007132967672682259, −10.92566272290476703471156793278, −10.4853581392265763269165685305, −9.68767933409311342087674252395, −9.29202810396633564743628077775, −8.2703921617786962832702035483, −7.80920140009429602145446675065, −7.26742943149937242308328649208, −6.74289064745608383623320477295, −5.20038565737373568467875796122, −4.35048147219936200856209025459, −3.53961435316904555099921311119, −2.97063142235214484458097701756, −2.2856058685145533395468780554, −1.102955926381855324678167762490, −0.2049744849868935383856843393, 1.06244563128141430322250773840, 2.259789183025626602878144064865, 2.785055707916513810697933202377, 3.44040839634096623127493795220, 4.54340476269529840714352086889, 5.48605079501721972883815878956, 6.55778692197112228768328565889, 7.020891854435737773363398033085, 7.743306597211993464690785216937, 8.40004357726405441003488100628, 8.960537366836412768464703889021, 9.53341611342798790734087226443, 10.353073751457323050501739808601, 11.1347918043959252868827146889, 11.758044270080710379092258257144, 12.72838816209156811265695508244, 13.23119843705190828204335654333, 14.3766199989008483552407300941, 14.99172960747568741351634541923, 15.31321221815219263493678915710, 16.12524114166691039416517640585, 16.4374789615753159867752548725, 17.58091343176437582685361698827, 18.38001554351464340263975915859, 18.88888263037963707195528048779

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