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Properties

Label 1-4033-4033.969-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.896 + 0.443i$
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 + (0.973 + 0.230i)3-s + 4-s + (−0.0581 + 0.998i)5-s + (0.973 + 0.230i)6-s + (0.893 − 0.448i)7-s + 8-s + (0.893 + 0.448i)9-s + (−0.0581 + 0.998i)10-s + (−0.0581 − 0.998i)11-s + (0.973 + 0.230i)12-s + (−0.0581 + 0.998i)13-s + (0.893 − 0.448i)14-s + (−0.286 + 0.957i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + 2-s + (0.973 + 0.230i)3-s + 4-s + (−0.0581 + 0.998i)5-s + (0.973 + 0.230i)6-s + (0.893 − 0.448i)7-s + 8-s + (0.893 + 0.448i)9-s + (−0.0581 + 0.998i)10-s + (−0.0581 − 0.998i)11-s + (0.973 + 0.230i)12-s + (−0.0581 + 0.998i)13-s + (0.893 − 0.448i)14-s + (−0.286 + 0.957i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.896 + 0.443i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (969, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.896 + 0.443i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(6.070395992 + 1.418559962i\)
\(L(\frac12)\) \(\approx\) \(6.070395992 + 1.418559962i\)
\(L(1)\) \(\approx\) \(3.049744531 + 0.4719719309i\)
\(L(1)\) \(\approx\) \(3.049744531 + 0.4719719309i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (0.973 + 0.230i)T \)
5 \( 1 + (-0.0581 + 0.998i)T \)
7 \( 1 + (0.893 - 0.448i)T \)
11 \( 1 + (-0.0581 - 0.998i)T \)
13 \( 1 + (-0.0581 + 0.998i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.893 - 0.448i)T \)
31 \( 1 + (-0.993 - 0.116i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (0.973 - 0.230i)T \)
53 \( 1 + (-0.686 + 0.727i)T \)
59 \( 1 + (-0.286 + 0.957i)T \)
61 \( 1 + (0.597 + 0.802i)T \)
67 \( 1 + (0.973 - 0.230i)T \)
71 \( 1 + T \)
73 \( 1 + (0.893 + 0.448i)T \)
79 \( 1 + (-0.286 + 0.957i)T \)
83 \( 1 + (0.396 - 0.918i)T \)
89 \( 1 + (-0.835 - 0.549i)T \)
97 \( 1 + (-0.993 + 0.116i)T \)
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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

−18.41306601673670346581556572499, −17.685212713946798181637602540043, −17.10684299976518006424626479686, −15.85136242119361223714725729816, −15.64286266122262806591268904388, −14.94188474124386439905987497854, −14.27044054882608482259908893550, −13.73495511886448869774691021593, −12.831669820398845368346487214596, −12.42563859592130458774960252564, −12.04500028819709494287518443868, −10.93929222359864151099562211033, −10.12211891915891254289113239603, −9.38524205875302093583764435060, −8.36097878474948918662862975588, −7.99417050946512031414758768932, −7.374347687895114059571496853712, −6.37680206217626273686326408850, −5.353094925383151932106604908602, −5.01274101306165232786007465880, −4.03111265945209051995732379385, −3.58925136533940080061594371143, −2.37725543209014712214352464076, −1.86686010699173917723396654154, −1.20638692747490564359168859658, 1.19514051204266202677916080382, 2.24218558768745144508478294085, 2.66123772010321654214728811448, 3.53119659093775183741548731123, 4.16547201582828205323160532169, 4.78090379304940312810291657272, 5.74734815978713393326125556109, 6.72096762116820303003464623318, 7.22237904485287417461047942381, 7.86925173857364413845603732706, 8.64960339894756934217578711636, 9.639186454440806787770091435393, 10.389407297265272144286731492854, 11.27180360582825285419780528768, 11.36869443402552877927965069429, 12.4083870395739274640017801288, 13.65099000463321711409631035064, 13.86205932181563114474254037121, 14.15415894091358525854940786179, 14.93669967965982661733233653765, 15.63252596522017636266910787507, 16.14256219217177162034066012513, 16.93043316740447141881129783662, 18.08706498317027840702674861717, 18.56806379742841005654406894057

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