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Properties

Label 1-4033-4033.3617-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.731 - 0.682i$
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 + (0.973 + 0.230i)3-s + (−0.939 + 0.342i)4-s + (−0.893 + 0.448i)5-s + (−0.0581 + 0.998i)6-s + (−0.835 − 0.549i)7-s + (−0.5 − 0.866i)8-s + (0.893 + 0.448i)9-s + (−0.597 − 0.802i)10-s + (−0.597 + 0.802i)11-s + (−0.993 + 0.116i)12-s + (−0.0581 + 0.998i)13-s + (0.396 − 0.918i)14-s + (−0.973 + 0.230i)15-s + (0.766 − 0.642i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (0.973 + 0.230i)3-s + (−0.939 + 0.342i)4-s + (−0.893 + 0.448i)5-s + (−0.0581 + 0.998i)6-s + (−0.835 − 0.549i)7-s + (−0.5 − 0.866i)8-s + (0.893 + 0.448i)9-s + (−0.597 − 0.802i)10-s + (−0.597 + 0.802i)11-s + (−0.993 + 0.116i)12-s + (−0.0581 + 0.998i)13-s + (0.396 − 0.918i)14-s + (−0.973 + 0.230i)15-s + (0.766 − 0.642i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.731 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.731 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.731 - 0.682i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3617, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.731 - 0.682i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5168296966 + 1.311347947i\)
\(L(\frac12)\) \(\approx\) \(-0.5168296966 + 1.311347947i\)
\(L(1)\) \(\approx\) \(0.6878011025 + 0.8482763265i\)
\(L(1)\) \(\approx\) \(0.6878011025 + 0.8482763265i\)

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 \( 1 + (0.973 + 0.230i)T \)
5 \( 1 + (-0.893 + 0.448i)T \)
7 \( 1 + (-0.835 - 0.549i)T \)
11 \( 1 + (-0.597 + 0.802i)T \)
13 \( 1 + (-0.0581 + 0.998i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (-0.597 - 0.802i)T \)
31 \( 1 + (0.686 + 0.727i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (-0.893 - 0.448i)T \)
53 \( 1 + (0.993 - 0.116i)T \)
59 \( 1 + (0.973 - 0.230i)T \)
61 \( 1 + (0.993 - 0.116i)T \)
67 \( 1 + (-0.597 + 0.802i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (-0.993 - 0.116i)T \)
79 \( 1 + (0.396 + 0.918i)T \)
83 \( 1 + (-0.0581 + 0.998i)T \)
89 \( 1 + (-0.973 - 0.230i)T \)
97 \( 1 + (0.286 + 0.957i)T \)
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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

−18.51593173355724514137511570734, −17.75079490447023965403850406204, −16.39236235851172900257747267175, −15.9749314480975212150906818923, −15.21062763462116105847805563057, −14.59760896118027763261046537031, −13.67729745338927056734414963496, −13.15192414972307528142867545827, −12.62593586118165125690302869275, −12.020287438330015970495419868814, −11.310635841303424192023898375801, −10.37212198942340507129792983460, −9.734387034396145490166076155739, −8.97077426408311563335495643578, −8.50864818901566037368025102883, −7.80120272238676972507344077733, −7.038573518332392465006388154169, −5.754283362213333034095058512251, −5.16598005036729206542664587701, −4.2070547309659497445406618895, −3.34845557272028664273783483564, −2.99241022869337572152874462797, −2.37908817791916027552872174458, −1.00939462494186222057083486600, −0.404160837124159751171219861821, 1.237071145851322826804715243461, 2.5775322212010376889649976111, 3.37771959651590851331603439145, 3.96170227859737792068593446232, 4.46145208843726429286672149125, 5.440875206326556963746759304716, 6.620076214047430470954554432441, 7.04331472247483028080534858591, 7.720950810333790948323880834056, 8.15573828676403340147383138874, 9.13697867987431558446176870870, 9.79469151192077209153861944901, 10.22290898662205797574742136743, 11.3864180585695352062010204491, 12.30495660426962192722510895173, 13.031873207291415829770909383332, 13.55945187197844637331134403892, 14.28821595614722834139660302302, 14.94896905732041728448536759316, 15.38590345084717759515152414166, 16.19843197317836452321997409944, 16.35860714358517780551032968187, 17.503258773804435349309500853949, 18.19492467242561562032694675433, 19.08964252073316283803650236430

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