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Properties

Label 1-4033-4033.363-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.192 - 0.981i$
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  + (−0.939 + 0.342i)2-s + (0.893 + 0.448i)3-s + (0.766 − 0.642i)4-s + (−0.597 + 0.802i)5-s + (−0.993 − 0.116i)6-s + (0.396 + 0.918i)7-s + (−0.5 + 0.866i)8-s + (0.597 + 0.802i)9-s + (0.286 − 0.957i)10-s + (0.286 + 0.957i)11-s + (0.973 − 0.230i)12-s + (−0.993 − 0.116i)13-s + (−0.686 − 0.727i)14-s + (−0.893 + 0.448i)15-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.893 + 0.448i)3-s + (0.766 − 0.642i)4-s + (−0.597 + 0.802i)5-s + (−0.993 − 0.116i)6-s + (0.396 + 0.918i)7-s + (−0.5 + 0.866i)8-s + (0.597 + 0.802i)9-s + (0.286 − 0.957i)10-s + (0.286 + 0.957i)11-s + (0.973 − 0.230i)12-s + (−0.993 − 0.116i)13-s + (−0.686 − 0.727i)14-s + (−0.893 + 0.448i)15-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.192 - 0.981i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.192 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2694480206 + 0.3275125275i\)
\(L(\frac12)\) \(\approx\) \(-0.2694480206 + 0.3275125275i\)
\(L(1)\) \(\approx\) \(0.5839735724 + 0.4575477609i\)
\(L(1)\) \(\approx\) \(0.5839735724 + 0.4575477609i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.939 + 0.342i)T \)
3 \( 1 + (0.893 + 0.448i)T \)
5 \( 1 + (-0.597 + 0.802i)T \)
7 \( 1 + (0.396 + 0.918i)T \)
11 \( 1 + (0.286 + 0.957i)T \)
13 \( 1 + (-0.993 - 0.116i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (0.286 - 0.957i)T \)
31 \( 1 + (0.0581 - 0.998i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (-0.597 - 0.802i)T \)
53 \( 1 + (-0.973 + 0.230i)T \)
59 \( 1 + (0.893 - 0.448i)T \)
61 \( 1 + (-0.973 + 0.230i)T \)
67 \( 1 + (0.286 + 0.957i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (0.973 + 0.230i)T \)
79 \( 1 + (-0.686 + 0.727i)T \)
83 \( 1 + (-0.993 - 0.116i)T \)
89 \( 1 + (-0.893 - 0.448i)T \)
97 \( 1 + (0.835 - 0.549i)T \)
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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

−17.97780240547298828442364816375, −17.48391671613231202095415101712, −16.60373858204057855532849641395, −16.11271269701368618820901634456, −15.49641003619386047535951307234, −14.42294320216600232863612024549, −13.923304735403372025251628297065, −13.098440361714004116894546350387, −12.39276966258643859021561569810, −11.78086418473555288150931875605, −11.12918510514513432676496855756, −10.24236061590555059328586973333, −9.462978262413188932162183841488, −8.87918564159215608817365315104, −8.250961599389809254900151108326, −7.690726026211152778908026562842, −7.10909545349389569268238291476, −6.40521527495703068556706900580, −4.989680371649803471302100861377, −4.18481715588519394628667316458, −3.42846071394688004911790403273, −2.7565490422153315164138828054, −1.63135213703800369373866841254, −1.110217927149748016680014310519, −0.14213366969730680525780356788, 1.68308187111406552264664848304, 2.334847516610992352774053083531, 2.79625083658198658647913899544, 3.96220651167735501356310185859, 4.724918261379237242217117768715, 5.64173208396047439863115733298, 6.608297063443333372361099132920, 7.37762899818052352035915522870, 7.83572572835232264998091753232, 8.46071247897584296052630815653, 9.33350830759249909574959258169, 9.82506195145430719767066163687, 10.36758048619610363914921103556, 11.4311974469102809670585693588, 11.76148575745269317368382726267, 12.68087069610249484443204778519, 13.94252675078936625936208010566, 14.48414558596846009732363011569, 15.11385441198364982007292726565, 15.50235845550052450589814381854, 15.903286212235239697333239777990, 17.03400194146831161454602770696, 17.75049366251935092101973720330, 18.272514799949941099828911291187, 19.01236010489952903169280480603

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