Properties

Label 1-4033-4033.3460-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.119 + 0.992i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  i·2-s + (−0.893 + 0.448i)3-s − 4-s + (−0.116 − 0.993i)5-s + (0.448 + 0.893i)6-s + (0.597 + 0.802i)7-s + i·8-s + (0.597 − 0.802i)9-s + (−0.993 + 0.116i)10-s + (0.993 + 0.116i)11-s + (0.893 − 0.448i)12-s + (−0.116 − 0.993i)13-s + (0.802 − 0.597i)14-s + (0.549 + 0.835i)15-s + 16-s + (0.866 − 0.5i)17-s + ⋯
L(s)  = 1  i·2-s + (−0.893 + 0.448i)3-s − 4-s + (−0.116 − 0.993i)5-s + (0.448 + 0.893i)6-s + (0.597 + 0.802i)7-s + i·8-s + (0.597 − 0.802i)9-s + (−0.993 + 0.116i)10-s + (0.993 + 0.116i)11-s + (0.893 − 0.448i)12-s + (−0.116 − 0.993i)13-s + (0.802 − 0.597i)14-s + (0.549 + 0.835i)15-s + 16-s + (0.866 − 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.119 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.119 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.119 + 0.992i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3460, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.119 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01406189475 + 0.01586218538i\)
\(L(\frac12)\) \(\approx\) \(0.01406189475 + 0.01586218538i\)
\(L(1)\) \(\approx\) \(0.6598346476 - 0.3586095432i\)
\(L(1)\) \(\approx\) \(0.6598346476 - 0.3586095432i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - iT \)
3 \( 1 + (-0.893 + 0.448i)T \)
5 \( 1 + (-0.116 - 0.993i)T \)
7 \( 1 + (0.597 + 0.802i)T \)
11 \( 1 + (0.993 + 0.116i)T \)
13 \( 1 + (-0.116 - 0.993i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (-0.984 + 0.173i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (0.802 - 0.597i)T \)
31 \( 1 + (-0.230 - 0.973i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (0.893 + 0.448i)T \)
53 \( 1 + (-0.0581 + 0.998i)T \)
59 \( 1 + (-0.549 - 0.835i)T \)
61 \( 1 + (-0.957 + 0.286i)T \)
67 \( 1 + (-0.893 - 0.448i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.597 + 0.802i)T \)
79 \( 1 + (-0.549 - 0.835i)T \)
83 \( 1 + (-0.686 + 0.727i)T \)
89 \( 1 + (-0.918 - 0.396i)T \)
97 \( 1 + (-0.230 + 0.973i)T \)
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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

−18.60663936520087880181102965699, −17.95576501432703683315643631794, −17.31282263080207319446166700326, −16.76509125613602516502432454397, −16.43071818348367878826998767530, −15.33819287836704193712310507596, −14.60648894465864979429696207597, −14.17476058664431002168648692622, −13.64107627614311023677704329654, −12.603083849237647019332873013, −12.023857881181272866646639668984, −11.13924155179333319187902675365, −10.55812147754155127605683871313, −9.97210250392686551359789665886, −8.84156920929053947386043366755, −8.165546419863273900670078865317, −7.190092720727614781108323883454, −6.92084361459038327117522026362, −6.37696841637609763632673233075, −5.61079964610217563185824878945, −4.52653516994632762200013640426, −4.23533648499906411163215358205, −3.199882101421999218275684536301, −1.75942803862094021558921647638, −1.06830182924264063458539310656, 0.004560694247810970017119876531, 0.940586631175613342830226819627, 1.42399222777040579699831564027, 2.51711724297641617873327927121, 3.55883073128706234359694396302, 4.32056120915443639825536919904, 4.859568344230850792872936075912, 5.61399955115890557149611756102, 6.0135778997977493868034713922, 7.50610716952958010314650401968, 8.25076948972710622089561767604, 9.12809653266104122993103435903, 9.42588428464001013331386758954, 10.30219285994849697823630374983, 11.04727602443048109361638257026, 11.71142736621868765634903741344, 12.25698578302001159104510454932, 12.54866022384923846108451924952, 13.462413280097936161874517623612, 14.39248390038766816707168311026, 15.17140827538871481370042807438, 15.68798241619143963780634838541, 16.80726693492645899869491547083, 17.27314104681980512069173169526, 17.58282534070875075284283402456

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