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Properties

Label 1-4033-4033.3112-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: $\chi_{4033} (3112, \cdot )$
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

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

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