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Properties

Label 1-4033-4033.2625-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.966 - 0.257i$
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  + (−0.866 + 0.5i)2-s + (0.286 − 0.957i)3-s + (0.5 − 0.866i)4-s + (−0.448 + 0.893i)5-s + (0.230 + 0.973i)6-s + (−0.835 + 0.549i)7-s + i·8-s + (−0.835 − 0.549i)9-s + (−0.0581 − 0.998i)10-s + (0.0581 − 0.998i)11-s + (−0.686 − 0.727i)12-s + (−0.448 + 0.893i)13-s + (0.448 − 0.893i)14-s + (0.727 + 0.686i)15-s + (−0.5 − 0.866i)16-s + i·17-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.286 − 0.957i)3-s + (0.5 − 0.866i)4-s + (−0.448 + 0.893i)5-s + (0.230 + 0.973i)6-s + (−0.835 + 0.549i)7-s + i·8-s + (−0.835 − 0.549i)9-s + (−0.0581 − 0.998i)10-s + (0.0581 − 0.998i)11-s + (−0.686 − 0.727i)12-s + (−0.448 + 0.893i)13-s + (0.448 − 0.893i)14-s + (0.727 + 0.686i)15-s + (−0.5 − 0.866i)16-s + i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.966 - 0.257i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.966 - 0.257i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0003154127403 + 0.002409585856i\)
\(L(\frac12)\) \(\approx\) \(0.0003154127403 + 0.002409585856i\)
\(L(1)\) \(\approx\) \(0.5601083452 + 0.03277251858i\)
\(L(1)\) \(\approx\) \(0.5601083452 + 0.03277251858i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.286 - 0.957i)T \)
5 \( 1 + (-0.448 + 0.893i)T \)
7 \( 1 + (-0.835 + 0.549i)T \)
11 \( 1 + (0.0581 - 0.998i)T \)
13 \( 1 + (-0.448 + 0.893i)T \)
17 \( 1 + iT \)
19 \( 1 + (-0.342 - 0.939i)T \)
23 \( 1 + (-0.342 - 0.939i)T \)
29 \( 1 + (0.448 - 0.893i)T \)
31 \( 1 + (0.116 + 0.993i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (0.642 + 0.766i)T \)
47 \( 1 + (0.973 + 0.230i)T \)
53 \( 1 + (-0.286 + 0.957i)T \)
59 \( 1 + (-0.727 - 0.686i)T \)
61 \( 1 + (0.918 - 0.396i)T \)
67 \( 1 + (0.686 - 0.727i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.893 + 0.448i)T \)
79 \( 1 + (-0.230 + 0.973i)T \)
83 \( 1 + (0.597 - 0.802i)T \)
89 \( 1 + (0.448 - 0.893i)T \)
97 \( 1 + (0.116 - 0.993i)T \)
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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

−17.684755081617597127014470324957, −17.23940270886202756133703269916, −16.564290339504342323638252934, −16.01474990984589351491956037956, −15.53415273414156452488891281865, −14.7971842692332576439787918945, −13.64786691329421893136712311977, −13.03551466287880164326439610669, −12.16405618015686222658112348420, −11.86286932709828689145472325027, −10.69631969841506014059426383764, −10.22088335636644138046159763994, −9.57635841803156467036527401676, −9.18211052426661782404932799233, −8.29961954245984544596095841951, −7.61354877009762914439310931738, −7.09310590097369936587681874274, −5.79479243315385813871427029402, −4.95791929650206642603607149528, −4.07911139430562221412850595650, −3.61647474807133275678534421220, −2.78143883953413539081404059585, −1.869167451054497929134865168707, −0.70693091079075262164289122391, −0.000789073225883305844604336829, 0.79645737924167242258132171949, 1.98703833481197000456957747820, 2.62486928981979914499519324692, 3.24522580439118497294252892353, 4.44248714917442335986212122006, 5.81084862547032666660126360639, 6.368772295447491988262481962352, 6.6758266296751852651770213221, 7.44470398693581881969606376510, 8.31685796842509452028467491827, 8.68820399552825871981510106592, 9.484706531670258645756321920000, 10.34542991503812980744845607730, 11.09371982165921422169613976313, 11.71555725322362170893587074911, 12.39384748067280834961280154704, 13.30003398761747743942558428732, 14.23480503692972538099132457275, 14.45190401091239183149155301100, 15.45732410705115271207843960150, 15.8399632728807116357489862438, 16.81420216003058852551539286580, 17.330170362717690033744642888492, 18.198511819520090476197396631261, 18.8059371801392480377411081687

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