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Properties

Label 1-4033-4033.368-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.985 - 0.168i$
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 + (−0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (0.342 − 0.939i)5-s + (−0.5 − 0.866i)6-s + (−0.939 − 0.342i)7-s + (0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (0.866 + 0.5i)10-s + (−0.866 + 0.5i)11-s + (0.939 − 0.342i)12-s + (−0.173 − 0.984i)13-s + (0.5 − 0.866i)14-s + (0.342 + 0.939i)15-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (0.342 − 0.939i)5-s + (−0.5 − 0.866i)6-s + (−0.939 − 0.342i)7-s + (0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (0.866 + 0.5i)10-s + (−0.866 + 0.5i)11-s + (0.939 − 0.342i)12-s + (−0.173 − 0.984i)13-s + (0.5 − 0.866i)14-s + (0.342 + 0.939i)15-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.985 - 0.168i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.985 - 0.168i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02163004570 - 0.2546072379i\)
\(L(\frac12)\) \(\approx\) \(0.02163004570 - 0.2546072379i\)
\(L(1)\) \(\approx\) \(0.4924750542 + 0.06437925501i\)
\(L(1)\) \(\approx\) \(0.4924750542 + 0.06437925501i\)

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

−18.60068276765520181077378488201, −18.42737246387100569554143896631, −17.70599585948990960070032829174, −16.96547488635177533552123505600, −16.21938123158354048552759179303, −15.56939911223967919698256007681, −14.28559294544972592663392626185, −13.86102488512422280441171345647, −13.057372957467325701239873241712, −12.71809354623038793073233566673, −11.72045966711541703719148453359, −11.288968067873698267400965852, −10.70705798924649054429160404992, −9.829664163256848450647881617117, −9.51146926839027739407696189573, −8.348559454571097877900335804742, −7.61582399316995715107187782206, −6.80855928706183842638910460067, −6.05221817186028568859887895720, −5.52925194211926798066976037792, −4.46625091905547120744357916674, −3.58259732899015561071508071225, −2.65271364035266605506142225681, −2.222429775557890746053413843675, −1.31511368405173351008096383878, 0.16614445576654755502859467088, 0.51396133986585103204859215446, 2.003862721909482315160487494, 3.32446081124732717396295432328, 4.23500625126394220744682365532, 4.819233196236287794858731718299, 5.439960296957165654839770904600, 6.055495055186051262050064942371, 6.82176405370241020701172343640, 7.49355196210878587066870092581, 8.590605448224962651043984959652, 9.04259563866363285961949371190, 9.83689445853792364167836623493, 10.3519653957058823971156261853, 10.918828093711258728243502424071, 12.3529824012580129200769266019, 12.80551126170901420407503339588, 13.19340290698978606937822221875, 14.11947370892069457106630237390, 15.28615247652205773751562459675, 15.46530304020095204423856263440, 16.18254301233900486381337670125, 16.71125356699428293525431087740, 17.32623350466226553957225649028, 17.87498917343760229171032781161

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