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Properties

Label 1-4033-4033.2235-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.585 + 0.810i$
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 + (0.939 − 0.342i)3-s + (−0.939 − 0.342i)4-s + (0.342 + 0.939i)5-s + (0.173 + 0.984i)6-s + (−0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.984 + 0.173i)10-s + (0.984 + 0.173i)11-s − 12-s + (−0.766 − 0.642i)13-s + (−0.173 − 0.984i)14-s + (0.642 + 0.766i)15-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.939 − 0.342i)3-s + (−0.939 − 0.342i)4-s + (0.342 + 0.939i)5-s + (0.173 + 0.984i)6-s + (−0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.984 + 0.173i)10-s + (0.984 + 0.173i)11-s − 12-s + (−0.766 − 0.642i)13-s + (−0.173 − 0.984i)14-s + (0.642 + 0.766i)15-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.585 + 0.810i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.585 + 0.810i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.837473350 + 0.9389550357i\)
\(L(\frac12)\) \(\approx\) \(1.837473350 + 0.9389550357i\)
\(L(1)\) \(\approx\) \(1.145373944 + 0.5283467866i\)
\(L(1)\) \(\approx\) \(1.145373944 + 0.5283467866i\)

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

−18.724288578004344352862655257568, −17.59424623334998426536363014020, −17.13960024229406533540389272847, −16.286111691204238935070647332187, −15.90138448528661446621810422292, −14.69377086302138888316368227782, −13.95879061957763915119655902501, −13.660314839397178200208755586369, −12.69780718897547589251119727485, −12.52709150292293170701651740419, −11.40770784738674996690360181253, −10.76835477135780803340488123742, −9.670152125543773321955858328268, −9.28456048094250642598430048080, −9.21130941706758921746095440430, −8.21918269782346137341071605929, −7.30940211975829863267310648541, −6.55226252199407190934430594199, −5.217655546503382219217303824014, −4.5832291361110933220811939878, −4.005829003838693668614605843882, −3.144873428175073577096436595068, −2.54077916740766146580289115160, −1.56669177854823005799152741302, −0.86865718433133192131174167513, 0.69695277727582542130841222021, 1.94011673397918451737813424994, 2.74694886659889428542515407411, 3.53604877709810089170088295622, 4.183355864271603737790092957548, 5.37648836492395169722948680698, 6.2486964304641803472990665561, 6.6563048110535237284516407169, 7.3665792180093063988811925799, 7.88412539486739934887285440667, 8.96413244850659832439775582977, 9.44493821388778526379739421334, 9.84720686430180280982904092094, 10.72555838392476424095101518456, 11.91141833074256594566258995230, 12.72531760407081140433516015861, 13.379532260653321116616262447107, 13.883254697363354251329204516737, 14.67002835340025673040017611098, 15.16671572778737338298341987074, 15.50665226659067483923801741138, 16.51538294113851037692541484777, 17.35690346756371187730020295480, 17.81435517117948313073431574092, 18.61175522707668852208555197098

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