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Properties

Label 1-4033-4033.3283-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.485 + 0.874i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (−0.984 − 0.173i)2-s + (−0.835 − 0.549i)3-s + (0.939 + 0.342i)4-s + (0.993 − 0.116i)5-s + (0.727 + 0.686i)6-s + (0.597 + 0.802i)7-s + (−0.866 − 0.5i)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 + (−0.802 + 0.597i)13-s + (−0.448 − 0.893i)14-s + (−0.893 − 0.448i)15-s + (0.766 + 0.642i)16-s + (0.642 − 0.766i)17-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)2-s + (−0.835 − 0.549i)3-s + (0.939 + 0.342i)4-s + (0.993 − 0.116i)5-s + (0.727 + 0.686i)6-s + (0.597 + 0.802i)7-s + (−0.866 − 0.5i)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 + (−0.802 + 0.597i)13-s + (−0.448 − 0.893i)14-s + (−0.893 − 0.448i)15-s + (0.766 + 0.642i)16-s + (0.642 − 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.485 + 0.874i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.485 + 0.874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4024886344 + 0.6841127226i\)
\(L(\frac12)\) \(\approx\) \(0.4024886344 + 0.6841127226i\)
\(L(1)\) \(\approx\) \(0.6307259067 + 0.01284102874i\)
\(L(1)\) \(\approx\) \(0.6307259067 + 0.01284102874i\)

Euler product

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

−17.59177674886228224531990389339, −17.55176846035225618640577902508, −16.9188701617357614369815132772, −16.396753177053479907038063021488, −15.32693272182311955630193878265, −14.98742118423655579453720339151, −14.23556761355955407373091399513, −13.077256865557489641554598856152, −12.63471953710493434805433401926, −11.50731165479651720780141688209, −10.797213209594891283529218274035, −10.51835564730424302890632983591, −9.9132536531620740717201126988, −9.25767941822101271739321504724, −8.36208032614279300189411594103, −7.49208560562938512026709167717, −6.94153045824868422168748468759, −6.09397048726300090252153058349, −5.2913250945986186653348090558, −5.00988178023888851794847834405, −3.707414547874498976065090556545, −2.67914043464865273182131239079, −1.86019069888545647571114984941, −0.88418182462430013827518108779, −0.23325505488699915886291775120, 0.91459782332499266604822041487, 1.66957363987751293932318837727, 2.34107160561610903758220841008, 2.87427485938495016100965130316, 4.62052441413868324607449140226, 5.2976374458458233030939195564, 5.878916974953654703535059868701, 6.64614837966483813306602797561, 7.407398929146515146272094452362, 8.03468957164298950783813482106, 8.83822084452055830171937948617, 9.63585491712006138171631597951, 10.20742534226521315999782809772, 10.909771630636248472594391283432, 11.58684861905815513088761128035, 12.287208704973647817665193549917, 12.75142227639302989766058148260, 13.57464408687952145890813108634, 14.584007483729359692300683855194, 15.17828463301893832217968164621, 16.32973884808925612688895534392, 16.57500613233339128651669032226, 17.329890988126803280692461384865, 17.97251100598868529881373900487, 18.33190442837475349824319071491

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