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Properties

Label 1-4033-4033.2058-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.914 - 0.404i$
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 + (0.835 + 0.549i)3-s + (−0.939 − 0.342i)4-s + (−0.116 − 0.993i)5-s + (−0.686 + 0.727i)6-s + (0.597 + 0.802i)7-s + (0.5 − 0.866i)8-s + (0.396 + 0.918i)9-s + (0.998 + 0.0581i)10-s + (−0.998 + 0.0581i)11-s + (−0.597 − 0.802i)12-s + (−0.597 − 0.802i)13-s + (−0.893 + 0.448i)14-s + (0.448 − 0.893i)15-s + (0.766 + 0.642i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.835 + 0.549i)3-s + (−0.939 − 0.342i)4-s + (−0.116 − 0.993i)5-s + (−0.686 + 0.727i)6-s + (0.597 + 0.802i)7-s + (0.5 − 0.866i)8-s + (0.396 + 0.918i)9-s + (0.998 + 0.0581i)10-s + (−0.998 + 0.0581i)11-s + (−0.597 − 0.802i)12-s + (−0.597 − 0.802i)13-s + (−0.893 + 0.448i)14-s + (0.448 − 0.893i)15-s + (0.766 + 0.642i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.914 - 0.404i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2058, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.914 - 0.404i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1898047105 + 0.8980871276i\)
\(L(\frac12)\) \(\approx\) \(-0.1898047105 + 0.8980871276i\)
\(L(1)\) \(\approx\) \(0.7938697638 + 0.5967541422i\)
\(L(1)\) \(\approx\) \(0.7938697638 + 0.5967541422i\)

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 \( 1 + (0.835 + 0.549i)T \)
5 \( 1 + (-0.116 - 0.993i)T \)
7 \( 1 + (0.597 + 0.802i)T \)
11 \( 1 + (-0.998 + 0.0581i)T \)
13 \( 1 + (-0.597 - 0.802i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.549 - 0.835i)T \)
31 \( 1 + (-0.918 + 0.396i)T \)
41 \( 1 + iT \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (0.957 + 0.286i)T \)
53 \( 1 + (-0.918 - 0.396i)T \)
59 \( 1 + (-0.893 - 0.448i)T \)
61 \( 1 + (-0.230 + 0.973i)T \)
67 \( 1 + (-0.116 + 0.993i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (0.0581 + 0.998i)T \)
79 \( 1 + (-0.597 + 0.802i)T \)
83 \( 1 + (0.835 + 0.549i)T \)
89 \( 1 + (0.230 - 0.973i)T \)
97 \( 1 + (-0.116 - 0.993i)T \)
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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

−18.38012944841626763002137291371, −17.752232637574267236098213143580, −16.96310552760572708541959541216, −16.12668516812648296260353642771, −14.89854928922329982053129954570, −14.44752127672408933066932232108, −13.99980823897774052484291894011, −13.39411220065744548791389330135, −12.41927541672231880078778940354, −12.09618942301331981514089439661, −10.92079885621070979692299370471, −10.66783629001997088849091091314, −9.88645552345084909864553616421, −9.156504323530898788953990045956, −8.27236626994492150165068662405, −7.60054455163351777011411292959, −7.27897233175128900988561057796, −6.25179427202869380124148177789, −5.02483966678705644034293498493, −4.231866067513159784221402683691, −3.53145717982531451491092866718, −2.75318906903439051336987940073, −2.20069876571972470132222843484, −1.425088607915755338731383577147, −0.24070513154651695813954872020, 1.21671986056809252418638741091, 2.188465708105612818163588306802, 3.1274084187194683851533731580, 4.170120165622615458014440921668, 4.81568474886009254996056398586, 5.38221401640927144625245283753, 5.87800305005924881426251901750, 7.37163663989379639839006950155, 7.8570793883442324942586207581, 8.35875745555571302585718549401, 8.93617078500609143180157760109, 9.62757031994902092872042120857, 10.2641073161756121279874903587, 11.103825742617017990142452663, 12.39977348557923638817531729478, 12.824195699754854181308539699997, 13.50825361316435169867938337044, 14.29156333140629304520618355353, 15.099857382693097904295550231522, 15.38915769344233407587283104553, 15.90909878728523798806951596865, 16.73930922111378420565081251324, 17.40427362777697799660111270523, 17.95798957530325425364156891511, 18.96983018402272286704951504510

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