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Properties

Label 1-4033-4033.2697-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.997 - 0.0755i$
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  + (−0.5 + 0.866i)2-s + (−0.835 − 0.549i)3-s + (−0.5 − 0.866i)4-s + (0.597 + 0.802i)5-s + (0.893 − 0.448i)6-s + (0.396 − 0.918i)7-s + 8-s + (0.396 + 0.918i)9-s + (−0.993 + 0.116i)10-s + (−0.993 − 0.116i)11-s + (−0.0581 + 0.998i)12-s + (0.597 + 0.802i)13-s + (0.597 + 0.802i)14-s + (−0.0581 − 0.998i)15-s + (−0.5 + 0.866i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.835 − 0.549i)3-s + (−0.5 − 0.866i)4-s + (0.597 + 0.802i)5-s + (0.893 − 0.448i)6-s + (0.396 − 0.918i)7-s + 8-s + (0.396 + 0.918i)9-s + (−0.993 + 0.116i)10-s + (−0.993 − 0.116i)11-s + (−0.0581 + 0.998i)12-s + (0.597 + 0.802i)13-s + (0.597 + 0.802i)14-s + (−0.0581 − 0.998i)15-s + (−0.5 + 0.866i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.997 - 0.0755i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.997 - 0.0755i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.208795958 - 0.04575729116i\)
\(L(\frac12)\) \(\approx\) \(1.208795958 - 0.04575729116i\)
\(L(1)\) \(\approx\) \(0.7855944564 + 0.1314405353i\)
\(L(1)\) \(\approx\) \(0.7855944564 + 0.1314405353i\)

Euler product

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

−18.38829355036294188789524907328, −17.77873751417060856779455200454, −17.35676098508300970304056244743, −16.57005126449287807333025410723, −15.89116420218668053832509205746, −15.422155030304501218289336653936, −14.29025901185563270104808273205, −13.35971643225832829201705082895, −12.77886638469593197334223183226, −12.14600603494602932250281149128, −11.66808731934966103490139701297, −10.83510366471314786281430942522, −10.11708613725474756395772994005, −9.70859436808440768091172010563, −8.97026966040824872867332799150, −8.15155108096881299883880438622, −7.679334612721638406695339239002, −6.19729184739561887408041799255, −5.50062596420974431409743198017, −5.08140853373077972275121039489, −4.31759055606618445822408963059, −3.20223642817978332337756370921, −2.61427866215579672184042385270, −1.34282238618589294972226083838, −0.95253855828322722612428710072, 0.62043442853596965129564758042, 1.34395712675686908390895759912, 2.25017703359565892257621612449, 3.41433052241241788993126119596, 4.69541272873694228376818846620, 5.12662667497625280963780242128, 6.00026938542211459717922789799, 6.57115709395299207617814809763, 7.29009453821261433792290732014, 7.610875719929949549671786032886, 8.549928262769054168747483930876, 9.52885049424460130736577626754, 10.3614517307944599859678410909, 10.715708140268198885299768505449, 11.251453951290464147166838309479, 12.33701124534494091133782976976, 13.326615703300489895516230465753, 13.880943772944590732958180684854, 14.12132010943341891220788845182, 15.200391820381813896479825898341, 15.94122444118548564249643082918, 16.59901204656378556362264833972, 17.138574954849753548255404397716, 17.82404776702560779684593465298, 18.20924636525782972079938325283

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