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Properties

Label 1-4033-4033.1958-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.964 + 0.263i$
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 + (−0.5 − 0.866i)3-s + (0.173 − 0.984i)4-s + (−0.5 + 0.866i)5-s + (−0.939 − 0.342i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.173 + 0.984i)10-s + (0.173 − 0.984i)11-s + (−0.939 + 0.342i)12-s + (−0.5 − 0.866i)13-s + (0.173 + 0.984i)14-s + 15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (−0.5 − 0.866i)3-s + (0.173 − 0.984i)4-s + (−0.5 + 0.866i)5-s + (−0.939 − 0.342i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.173 + 0.984i)10-s + (0.173 − 0.984i)11-s + (−0.939 + 0.342i)12-s + (−0.5 − 0.866i)13-s + (0.173 + 0.984i)14-s + 15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.964 + 0.263i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1958, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.964 + 0.263i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5607789205 + 0.07521027411i\)
\(L(\frac12)\) \(\approx\) \(0.5607789205 + 0.07521027411i\)
\(L(1)\) \(\approx\) \(0.7720870587 - 0.4644543246i\)
\(L(1)\) \(\approx\) \(0.7720870587 - 0.4644543246i\)

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 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.173 - 0.984i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (0.766 + 0.642i)T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (-0.939 + 0.342i)T \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (-0.939 - 0.342i)T \)
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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

−18.04448955188240711066324975614, −17.34347189157271061740339376223, −16.83464986729103762126077184968, −16.37884776132983812773459750557, −15.79630018785460237673025620404, −15.182752227596426201756673591793, −14.45609684840834710738464649109, −13.77064700241740391382426978932, −12.885096309315547676546760883173, −12.374613793972522722109885019104, −11.64847511956785291869588343220, −11.16367229417765093715340125353, −9.97822806303708947950916909601, −9.45539665093852573730163592397, −8.75310930673737319410333444061, −7.79324588822481524773724337021, −6.94423472099225629383553421521, −6.64530640302026114277310365582, −5.34300326848360676294966946874, −5.003332764282714770945700212202, −4.08706232163179746545143511330, −3.99304578788750776801816861654, −2.92441171316542726673131837651, −1.65850434158384440104032027540, −0.169672857240528476432267236087, 0.74102153646507346719574148098, 2.24292678655296078742255890947, 2.32566333808847614593823986279, 3.43712934451037315061634220551, 3.95830210613118520081865742546, 5.27909255659968884940695169030, 5.81986965406944468719957045473, 6.37061920599987300156419574029, 6.94587620599559689461511435526, 8.105047237793543978429326604782, 8.534320944281082840753534320016, 9.90678593933883258067566349955, 10.45527184589001315962019068713, 11.13050985875428056458940576043, 11.75212183475988576777190790175, 12.32532455159173332007127590218, 12.89922939752570334264497066633, 13.512006919729356440737168079325, 14.3875653285954089472446245436, 15.01354547909066058405345310487, 15.45347174631996551958818457708, 16.494606780365483962701910102636, 17.12696631372974406692211814721, 18.296636641984654168687177254744, 18.65008390382820845497779578484

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