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Properties

Label 1-4033-4033.1795-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.984 + 0.175i$
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  − 2-s + (0.835 + 0.549i)3-s + 4-s + (−0.802 + 0.597i)5-s + (−0.835 − 0.549i)6-s + (0.396 − 0.918i)7-s − 8-s + (0.396 + 0.918i)9-s + (0.802 − 0.597i)10-s + (−0.802 − 0.597i)11-s + (0.835 + 0.549i)12-s + (−0.597 − 0.802i)13-s + (−0.396 + 0.918i)14-s + (−0.998 + 0.0581i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (0.835 + 0.549i)3-s + 4-s + (−0.802 + 0.597i)5-s + (−0.835 − 0.549i)6-s + (0.396 − 0.918i)7-s − 8-s + (0.396 + 0.918i)9-s + (0.802 − 0.597i)10-s + (−0.802 − 0.597i)11-s + (0.835 + 0.549i)12-s + (−0.597 − 0.802i)13-s + (−0.396 + 0.918i)14-s + (−0.998 + 0.0581i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.984 + 0.175i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1795, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.984 + 0.175i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001247954195 + 0.01407986988i\)
\(L(\frac12)\) \(\approx\) \(0.001247954195 + 0.01407986988i\)
\(L(1)\) \(\approx\) \(0.6519819455 + 0.02805623044i\)
\(L(1)\) \(\approx\) \(0.6519819455 + 0.02805623044i\)

Euler product

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

−18.419062829869072433465367813263, −17.52239564768614367187931172756, −16.97685105531492509399528704789, −16.01910145435620743838707807785, −15.36420795519644892620831298950, −15.05202562065154057175581604498, −14.309367917430172199413062939857, −13.12852115643599947137017415320, −12.44499863172227295843565212604, −12.07081603836100198155209595090, −11.37545086295201149157864164131, −10.38839688825154191098308166927, −9.613034790834995022822367457685, −8.78362950905182182611126059362, −8.5789946021421135115069560836, −7.74658233453321421452581887837, −7.32332735388889942493747693387, −6.476111153173949728862277697281, −5.50268335161464449245759146482, −4.57727028820293074590929062764, −3.59013878284706825799635359830, −2.740415485995060459444088692562, −1.84473670139945922138382570212, −1.54031784635272117709362452016, −0.00532095265471664170095233108, 0.956310559579739601189907536500, 2.33186410529275547576243420877, 2.91369509163229932245680173911, 3.41078615063948601135344388489, 4.572519915813878588188155366616, 5.12840201298848528499029103913, 6.564036382285347271541540429064, 7.28665489667111748917385474978, 7.64970049095206206689153707122, 8.39950079315327179522548459266, 8.90379647989512290107398916008, 9.9587260992384848006557315344, 10.426518039408697301710681505286, 11.03711246802020787855258081512, 11.43418995449246721837875737209, 12.661459976756806082855355473956, 13.4089270928604456486101104205, 14.26301628581962939505609185583, 14.95677788748288932207153722981, 15.40104452217598771785474079953, 16.122561854426976730469501671379, 16.59062756039528485272940069608, 17.53730363605522706487087133172, 18.20741766144888347408274947965, 18.830934518260721923230483371527

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