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Properties

Label 1-4033-4033.1022-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.115 + 0.993i$
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  + (0.5 + 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s + i·5-s + (−0.5 − 0.866i)6-s + 7-s − 8-s + 9-s + (−0.866 + 0.5i)10-s + (0.866 + 0.5i)11-s + (0.5 − 0.866i)12-s − 13-s + (0.5 + 0.866i)14-s − i·15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s + i·5-s + (−0.5 − 0.866i)6-s + 7-s − 8-s + 9-s + (−0.866 + 0.5i)10-s + (0.866 + 0.5i)11-s + (0.5 − 0.866i)12-s − 13-s + (0.5 + 0.866i)14-s − i·15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.115 + 0.993i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1022, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.115 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.224240524 + 1.089632887i\)
\(L(\frac12)\) \(\approx\) \(1.224240524 + 1.089632887i\)
\(L(1)\) \(\approx\) \(0.8821456079 + 0.6235996929i\)
\(L(1)\) \(\approx\) \(0.8821456079 + 0.6235996929i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 - T \)
5 \( 1 + iT \)
7 \( 1 + T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 - T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 - T \)
61 \( 1 + iT \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−18.27761360051117724347111575109, −17.60606273962469422381834841279, −17.01587971778355120159945639993, −16.52673747141422941814116265522, −15.50180119496553363519035265875, −14.7880461097615734446376616688, −14.14758549057963100587164350262, −13.3070132475920723478731717220, −12.58342678105092435630059971991, −12.09375484575738772039442109567, −11.60677784340297737133254397796, −10.89999359889112408212628012660, −10.27312559981487046816693076751, −9.41610730685595831037580289228, −8.78378552949522468646724245386, −7.9528387414879251130604336699, −6.86717230665485641686834393672, −6.036179142389657614841550850156, −5.25047933328308991575676633382, −4.88066061095832234454703603818, −4.20014184642216975629420466988, −3.435880073138975377194772138423, −2.02707874839966807431845672785, −1.41660615811100037006739944675, −0.81652809050941534452501779496, 0.61369512296316966655420268230, 2.009833261942064048589162648841, 2.83888754204704760947005492600, 3.96386107595626811954339508331, 4.65350685447511078927314206395, 5.13471980856072013562621312091, 5.94463321268029372422408361244, 6.86892352996480991411680456617, 7.16512083871965497596491035127, 7.65394561946408750842191343063, 8.96145523998419893494440005032, 9.517991270268595270343097308234, 10.5354825072250011350310051822, 11.21978816461441832859514467576, 11.9847591404660039014056258177, 12.13314083213281463353626956575, 13.46319540327960795078028210505, 13.85325944380930699585084634553, 14.80744669400157911741651367472, 15.12910845056242435055691108199, 15.742042642078181162803771828129, 16.78793319593020543658144487133, 17.25250792042351450121808371751, 17.75475772557752222310561935195, 18.251722106294764563084354122335

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