Properties

Label 1-4033-4033.216-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.985 + 0.169i$
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 + (0.939 − 0.342i)3-s + (−0.5 − 0.866i)4-s + (0.642 − 0.766i)5-s + (0.173 − 0.984i)6-s + (0.766 + 0.642i)7-s − 8-s + (0.766 − 0.642i)9-s + (−0.342 − 0.939i)10-s + (0.342 − 0.939i)11-s + (−0.766 − 0.642i)12-s + (−0.766 − 0.642i)13-s + (0.939 − 0.342i)14-s + (0.342 − 0.939i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.939 − 0.342i)3-s + (−0.5 − 0.866i)4-s + (0.642 − 0.766i)5-s + (0.173 − 0.984i)6-s + (0.766 + 0.642i)7-s − 8-s + (0.766 − 0.642i)9-s + (−0.342 − 0.939i)10-s + (0.342 − 0.939i)11-s + (−0.766 − 0.642i)12-s + (−0.766 − 0.642i)13-s + (0.939 − 0.342i)14-s + (0.342 − 0.939i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.985 + 0.169i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (216, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.985 + 0.169i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2947370923 - 3.447794072i\)
\(L(\frac12)\) \(\approx\) \(-0.2947370923 - 3.447794072i\)
\(L(1)\) \(\approx\) \(1.214683794 - 1.569774020i\)
\(L(1)\) \(\approx\) \(1.214683794 - 1.569774020i\)

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

−18.57923292215830297220510218291, −17.990241858515283720821533155604, −17.48239847686867786123415446763, −16.68320186613382313286117802546, −16.013431494724522191195551766441, −15.13905847872625904284977079509, −14.61674936050321490353680566693, −14.254014787248351739707154361263, −13.67808472830746733847279439210, −13.11766203282421142747873585635, −12.01153590892673191437771559450, −11.40517392545351459251697486199, −10.27151660628872856699603670369, −9.60140488109144710920338380604, −9.28978749376929443477229471110, −8.025341798859192278249723929088, −7.66452790719571111744909179149, −6.94982659802384594825843034828, −6.42157486377731885379675210304, −5.05566281701420530630433301872, −4.81187213315640198291066457765, −3.85369714205284770865137793716, −3.25264704560052885460911566338, −2.213775275268286858385105449310, −1.66401134389701793101881445264, 0.63355961631646547020358183337, 1.50707082136068330041075140483, 2.17723329572568687192028157734, 2.75435775823659187608964007082, 3.66571965474614555752887156645, 4.52875678125947786812979393749, 5.20095830778289969232441777959, 5.897623810385273618216037805511, 6.719384157162032434227476735, 7.95784138303744794222318576457, 8.67416542547743112224755482626, 8.945078609955610607009586533255, 9.77190327881058250290054171927, 10.51039420536379052398677599911, 11.35927578397847290231286074713, 12.15233838977664127343090580567, 12.737549218433739969437406215235, 13.19318305851720197443414147553, 14.078986525472596927584714305995, 14.356466719817850605888506940829, 15.16702287593896307467997594386, 15.769814479712827360532824266323, 16.92399009609933027683471638482, 17.73499345218329190951581605758, 18.20889381576003536351785170709

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