Properties

Label 1-4033-4033.307-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.659 + 0.751i$
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  + (−0.766 − 0.642i)2-s + (0.597 + 0.802i)3-s + (0.173 + 0.984i)4-s + (0.686 − 0.727i)5-s + (0.0581 − 0.998i)6-s + (0.973 + 0.230i)7-s + (0.5 − 0.866i)8-s + (−0.286 + 0.957i)9-s + (−0.993 + 0.116i)10-s + (−0.993 − 0.116i)11-s + (−0.686 + 0.727i)12-s + (−0.973 − 0.230i)13-s + (−0.597 − 0.802i)14-s + (0.993 + 0.116i)15-s + (−0.939 + 0.342i)16-s + (0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.597 + 0.802i)3-s + (0.173 + 0.984i)4-s + (0.686 − 0.727i)5-s + (0.0581 − 0.998i)6-s + (0.973 + 0.230i)7-s + (0.5 − 0.866i)8-s + (−0.286 + 0.957i)9-s + (−0.993 + 0.116i)10-s + (−0.993 − 0.116i)11-s + (−0.686 + 0.727i)12-s + (−0.973 − 0.230i)13-s + (−0.597 − 0.802i)14-s + (0.993 + 0.116i)15-s + (−0.939 + 0.342i)16-s + (0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.659 + 0.751i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.659 + 0.751i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.414904668 + 0.6408248339i\)
\(L(\frac12)\) \(\approx\) \(1.414904668 + 0.6408248339i\)
\(L(1)\) \(\approx\) \(1.023092318 + 0.03537233947i\)
\(L(1)\) \(\approx\) \(1.023092318 + 0.03537233947i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 + (0.597 + 0.802i)T \)
5 \( 1 + (0.686 - 0.727i)T \)
7 \( 1 + (0.973 + 0.230i)T \)
11 \( 1 + (-0.993 - 0.116i)T \)
13 \( 1 + (-0.973 - 0.230i)T \)
17 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (-0.396 + 0.918i)T \)
31 \( 1 + (0.686 + 0.727i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (-0.835 + 0.549i)T \)
53 \( 1 + (0.973 + 0.230i)T \)
59 \( 1 + (0.993 + 0.116i)T \)
61 \( 1 + (0.0581 - 0.998i)T \)
67 \( 1 + (-0.286 + 0.957i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (-0.993 - 0.116i)T \)
79 \( 1 + (0.686 + 0.727i)T \)
83 \( 1 + (-0.993 + 0.116i)T \)
89 \( 1 + (0.835 + 0.549i)T \)
97 \( 1 + (-0.973 - 0.230i)T \)
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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

−18.45674154748237838083231876879, −17.67278760798809921726797164628, −17.1949139618087195679092403398, −16.71906799815700533927717853040, −15.35146330658624457810868725871, −14.77024333315797672074897987211, −14.63967626414383231998532912268, −13.709026304879001188915682930892, −13.24716252099284130304474321812, −12.15602755336536599470567950844, −11.42280467387576842589572561899, −10.41908846239986742272185582242, −10.141422375657147621575279824369, −9.27440142146155085502867242394, −8.34473162956043389917921687016, −7.93969664325344592489324024526, −7.20638582949505475278644003017, −6.78252388052632146734066981570, −5.739263638895781666983504433084, −5.3160754135368070270617154688, −4.16054773642596171408428261868, −2.865978169599534906160895027326, −2.127928924357946763845186644086, −1.73022940620149270390910998582, −0.53909761850134728953747674911, 1.00784024198874717073031615400, 1.880883415723847171436779587786, 2.65225349756694207389402282184, 3.12531454574648113977572637051, 4.420837896994647032769151693425, 4.97787529752728638286251814589, 5.42856205837010335874054718622, 6.923987263250479097856159346764, 7.85136916312674949090214672475, 8.27092816595327566871081399800, 8.927524789398373708287367393579, 9.60182877700519905389873842781, 10.15475282686133000684254434830, 10.8013383353510917501498151019, 11.475658870435153187274757539519, 12.397922738237946318532360924901, 13.03158833833822912905510201073, 13.66973881117724363031032025427, 14.58954193109601047949751331389, 15.14362446573554363893644595590, 16.082595385258966551364558871676, 16.595877302565640632339880036073, 17.2980509656955517363789519513, 17.80025843360364227104009896253, 18.580076419165040448054539357

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