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Properties

Label 1-4033-4033.3727-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.155 - 0.987i$
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 + (−0.686 − 0.727i)3-s + (−0.939 + 0.342i)4-s + (−0.893 − 0.448i)5-s + (−0.597 + 0.802i)6-s + (−0.835 + 0.549i)7-s + (0.5 + 0.866i)8-s + (−0.0581 + 0.998i)9-s + (−0.286 + 0.957i)10-s + (−0.286 − 0.957i)11-s + (0.893 + 0.448i)12-s + (0.835 − 0.549i)13-s + (0.686 + 0.727i)14-s + (0.286 + 0.957i)15-s + (0.766 − 0.642i)16-s + (−0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.686 − 0.727i)3-s + (−0.939 + 0.342i)4-s + (−0.893 − 0.448i)5-s + (−0.597 + 0.802i)6-s + (−0.835 + 0.549i)7-s + (0.5 + 0.866i)8-s + (−0.0581 + 0.998i)9-s + (−0.286 + 0.957i)10-s + (−0.286 − 0.957i)11-s + (0.893 + 0.448i)12-s + (0.835 − 0.549i)13-s + (0.686 + 0.727i)14-s + (0.286 + 0.957i)15-s + (0.766 − 0.642i)16-s + (−0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.155 - 0.987i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.155 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1798499793 - 0.2104152577i\)
\(L(\frac12)\) \(\approx\) \(0.1798499793 - 0.2104152577i\)
\(L(1)\) \(\approx\) \(0.3267018445 - 0.2948333618i\)
\(L(1)\) \(\approx\) \(0.3267018445 - 0.2948333618i\)

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 \( 1 + (-0.686 - 0.727i)T \)
5 \( 1 + (-0.893 - 0.448i)T \)
7 \( 1 + (-0.835 + 0.549i)T \)
11 \( 1 + (-0.286 - 0.957i)T \)
13 \( 1 + (0.835 - 0.549i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.973 + 0.230i)T \)
31 \( 1 + (-0.893 + 0.448i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (-0.993 + 0.116i)T \)
53 \( 1 + (-0.835 + 0.549i)T \)
59 \( 1 + (0.286 + 0.957i)T \)
61 \( 1 + (-0.597 + 0.802i)T \)
67 \( 1 + (-0.0581 + 0.998i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (-0.286 - 0.957i)T \)
79 \( 1 + (-0.893 + 0.448i)T \)
83 \( 1 + (-0.286 + 0.957i)T \)
89 \( 1 + (0.993 + 0.116i)T \)
97 \( 1 + (0.835 - 0.549i)T \)
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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

−18.518514632420811440944456891439, −17.87850985326962428553324303593, −17.107984684895331785106182710793, −16.433144025785175983652047813875, −15.87075006763737405872918809190, −15.62650239189914659029567567027, −14.75852288023905533601055413880, −14.206600176323365602116836300596, −13.18440236545180259738037375788, −12.619704679282301780395523337807, −11.63928300462978867051467834038, −11.0099780774394255117014249202, −10.21976479856291951216547977814, −9.64923452494677282720997000146, −9.068465011785091431625806002015, −7.99047111535596711904582658979, −7.37364637927111243861363226861, −6.62685807671168306803376923391, −6.199692649790315916068614231877, −5.25592163224761308713341992073, −4.356638336519003365004691056906, −3.92675344365711933924732019480, −3.32314475866275789405510866869, −1.690221975277932095703870378223, −0.27072467714751882419738194247, 0.361438628241130204042322519573, 1.28518186807899024248856141344, 2.27703515172515427013042979842, 3.11430601473646528667867284769, 3.8179144622143911635774145962, 4.691434398922389059830167424577, 5.58629728328155090078120315829, 6.13240236651268819307596570282, 7.1672515859257782601711646327, 7.995521557205540308984700178578, 8.70751887993339267682114037815, 9.01282672446738089895548467983, 10.329439831688429164554094627375, 10.97165008416181771729092628708, 11.30743125857794616580844345347, 12.18020271837530221869626265649, 12.71719825219136908146382613311, 13.16256708669486922415609426994, 13.70261169090289663851537686094, 14.909357697574223419480172270896, 15.86892901566106185990218303660, 16.282110537694958884996272794270, 17.009250127210886896233557954142, 17.89414740950298271651701567181, 18.4052766067793709725659814418

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