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Properties

Label 1-4033-4033.1070-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.911 - 0.412i$
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 + (−0.0581 + 0.998i)3-s + 4-s + (0.396 − 0.918i)5-s + (−0.0581 + 0.998i)6-s + (−0.993 + 0.116i)7-s + 8-s + (−0.993 − 0.116i)9-s + (0.396 − 0.918i)10-s + (0.396 + 0.918i)11-s + (−0.0581 + 0.998i)12-s + (0.396 − 0.918i)13-s + (−0.993 + 0.116i)14-s + (0.893 + 0.448i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + 2-s + (−0.0581 + 0.998i)3-s + 4-s + (0.396 − 0.918i)5-s + (−0.0581 + 0.998i)6-s + (−0.993 + 0.116i)7-s + 8-s + (−0.993 − 0.116i)9-s + (0.396 − 0.918i)10-s + (0.396 + 0.918i)11-s + (−0.0581 + 0.998i)12-s + (0.396 − 0.918i)13-s + (−0.993 + 0.116i)14-s + (0.893 + 0.448i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.911 - 0.412i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1070, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.911 - 0.412i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.077461328 - 0.6637147180i\)
\(L(\frac12)\) \(\approx\) \(3.077461328 - 0.6637147180i\)
\(L(1)\) \(\approx\) \(1.873753258 + 0.1083117533i\)
\(L(1)\) \(\approx\) \(1.873753258 + 0.1083117533i\)

Euler product

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

−18.86537389945439539190511533846, −17.76075242946076512139669192437, −17.159147442537727754842029119, −16.40939026755351675254886266156, −15.73288708578701912187899610569, −14.8707461094800708564838878733, −14.155335981533982071180547596189, −13.77066062704725523561823764693, −12.96695043373767801757019155079, −12.79976367596695320388813867003, −11.48873716963189214365671920182, −11.25783780473768639346859930310, −10.65175801835785001454430710950, −9.445860825169520869213802325436, −8.82029736445420251532649910461, −7.62192383158297533133075719478, −6.9882786383233334711723062489, −6.482964756796962534653406127979, −6.04283912778998178896386721121, −5.33484603601068522349008703293, −4.00175789877375640099992474455, −3.37060666180332282458591293293, −2.74308672178097913071130776566, −1.95832568406817664818286390068, −1.100806476289410034317823515858, 0.61630320082291428743990080715, 1.93531665985661247547039714397, 2.71856914138786390098799568059, 3.63566428171818782441979593377, 4.10068256002111217647764521140, 5.0139999174205922391587211703, 5.48300122370766104951406471200, 6.142315725785701543318389231006, 6.96899371239102448255287603863, 7.93827238367401367731567296693, 8.910134951992341950596421407175, 9.56515755530723510691389799610, 10.09292210715794574247622590222, 10.86567665798280179895353370934, 11.71129520245780377576451481869, 12.41182049491172000941891920762, 12.97376905022365093424261439760, 13.51800561369177875650751927478, 14.46152192370675769733625999330, 15.04978203022201528305573520433, 15.677277520037408493594767075031, 16.334072999166825980961255265711, 16.69142011987002272844579983936, 17.41734844854570443360181241614, 18.379730230402424473084038910020

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