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Properties

Label 1-4033-4033.2722-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: $\chi_{4033} (2722, \cdot )$
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

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

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