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Properties

Label 1-4033-4033.669-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.879 + 0.475i$
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  + (−0.173 + 0.984i)2-s + (−0.835 + 0.549i)3-s + (−0.939 − 0.342i)4-s + (−0.396 − 0.918i)5-s + (−0.396 − 0.918i)6-s + (−0.993 − 0.116i)7-s + (0.5 − 0.866i)8-s + (0.396 − 0.918i)9-s + (0.973 − 0.230i)10-s + (0.973 + 0.230i)11-s + (0.973 − 0.230i)12-s + (−0.597 + 0.802i)13-s + (0.286 − 0.957i)14-s + (0.835 + 0.549i)15-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.835 + 0.549i)3-s + (−0.939 − 0.342i)4-s + (−0.396 − 0.918i)5-s + (−0.396 − 0.918i)6-s + (−0.993 − 0.116i)7-s + (0.5 − 0.866i)8-s + (0.396 − 0.918i)9-s + (0.973 − 0.230i)10-s + (0.973 + 0.230i)11-s + (0.973 − 0.230i)12-s + (−0.597 + 0.802i)13-s + (0.286 − 0.957i)14-s + (0.835 + 0.549i)15-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.879 + 0.475i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (669, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.879 + 0.475i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1547414098 + 0.6116638192i\)
\(L(\frac12)\) \(\approx\) \(0.1547414098 + 0.6116638192i\)
\(L(1)\) \(\approx\) \(0.4921690065 + 0.3218553390i\)
\(L(1)\) \(\approx\) \(0.4921690065 + 0.3218553390i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 + (-0.835 + 0.549i)T \)
5 \( 1 + (-0.396 - 0.918i)T \)
7 \( 1 + (-0.993 - 0.116i)T \)
11 \( 1 + (0.973 + 0.230i)T \)
13 \( 1 + (-0.597 + 0.802i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.973 + 0.230i)T \)
31 \( 1 + (0.835 + 0.549i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (-0.993 + 0.116i)T \)
53 \( 1 + (0.973 - 0.230i)T \)
59 \( 1 + (0.835 + 0.549i)T \)
61 \( 1 + (0.286 - 0.957i)T \)
67 \( 1 + (-0.286 - 0.957i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (-0.686 + 0.727i)T \)
79 \( 1 + (0.686 - 0.727i)T \)
83 \( 1 + (-0.993 - 0.116i)T \)
89 \( 1 + (-0.893 - 0.448i)T \)
97 \( 1 + (-0.893 - 0.448i)T \)
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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

−18.203239419106028969062008663138, −17.76976311451217337066007194069, −17.013365912420999273812877638772, −16.38236641686020503965255981134, −15.47360511577309322417013064197, −14.72226832533750751703852502720, −13.706672388395992535142189707968, −13.29620124391175373364143411652, −12.55605668555984858580237804176, −11.7706456560870892208914363813, −11.44064856004527077600308325964, −10.820601747270980172832629096101, −9.89310699699100660230785301524, −9.54638740686370610319180778418, −8.53756369038541366079096475157, −7.39767655360121881493830867546, −7.1575181965724164487599396991, −6.21714061820428007435107339541, −5.43210028780115257191039867542, −4.56250845927511773072663551672, −3.65645414905360907534584714737, −2.8473215952996725123540012757, −2.41618192436387432712067544444, −1.05487829601673403164265101287, −0.36107213471909514476946983595, 0.77766034904275002358580088332, 1.5998222017546503782227201814, 3.47332825820420320032612310164, 4.05795874754128717751362185858, 4.575190174074584280789590369260, 5.456274646438190899465473697821, 6.04775951052120498241106656554, 6.829994048355133480705757407255, 7.2856342308794495971408949053, 8.475201679228965727566328059391, 9.08478589225031075467485767253, 9.624929379408753211725897270630, 10.177666474295294792356091594735, 11.18897829790840777923267453205, 12.11434607374187457398876636579, 12.55931763104181460234995157794, 13.183748761545463675954564291791, 14.252734426691175101268140997012, 14.86108951945086865929493356253, 15.60491431139653446998695881217, 16.19361555881941638094983208163, 16.762919477155453203474327791342, 17.02683678514520483883355933494, 17.66965805570789637339022727836, 18.76800655993053364595033304053

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