Properties

Label 1-4033-4033.324-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.703 - 0.710i$
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 + (−0.835 − 0.549i)3-s + (0.173 − 0.984i)4-s + (−0.396 + 0.918i)5-s + (−0.993 + 0.116i)6-s + (−0.993 + 0.116i)7-s + (−0.5 − 0.866i)8-s + (0.396 + 0.918i)9-s + (0.286 + 0.957i)10-s + (0.286 − 0.957i)11-s + (−0.686 + 0.727i)12-s + (0.597 + 0.802i)13-s + (−0.686 + 0.727i)14-s + (0.835 − 0.549i)15-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (−0.835 − 0.549i)3-s + (0.173 − 0.984i)4-s + (−0.396 + 0.918i)5-s + (−0.993 + 0.116i)6-s + (−0.993 + 0.116i)7-s + (−0.5 − 0.866i)8-s + (0.396 + 0.918i)9-s + (0.286 + 0.957i)10-s + (0.286 − 0.957i)11-s + (−0.686 + 0.727i)12-s + (0.597 + 0.802i)13-s + (−0.686 + 0.727i)14-s + (0.835 − 0.549i)15-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.703 - 0.710i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.703 - 0.710i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4832804062 - 1.158673025i\)
\(L(\frac12)\) \(\approx\) \(0.4832804062 - 1.158673025i\)
\(L(1)\) \(\approx\) \(0.8617576849 - 0.5148189136i\)
\(L(1)\) \(\approx\) \(0.8617576849 - 0.5148189136i\)

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 \( 1 + (-0.835 - 0.549i)T \)
5 \( 1 + (-0.396 + 0.918i)T \)
7 \( 1 + (-0.993 + 0.116i)T \)
11 \( 1 + (0.286 - 0.957i)T \)
13 \( 1 + (0.597 + 0.802i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.286 + 0.957i)T \)
31 \( 1 + (0.0581 + 0.998i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.597 + 0.802i)T \)
53 \( 1 + (0.686 - 0.727i)T \)
59 \( 1 + (-0.835 + 0.549i)T \)
61 \( 1 + (0.286 + 0.957i)T \)
67 \( 1 + (-0.973 + 0.230i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (0.973 - 0.230i)T \)
79 \( 1 + (-0.286 + 0.957i)T \)
83 \( 1 + (0.396 - 0.918i)T \)
89 \( 1 + (0.0581 - 0.998i)T \)
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

−18.549663777948826005860531683524, −17.626635964079362866479952466090, −17.103125271863029282294944676459, −16.570657141381688742274811915724, −16.01693646405796564469843823556, −15.38356001124579797982993117131, −15.013238401423073292253637174061, −13.9217899186275802651960470255, −13.06043174332465646914684342836, −12.43007681093855788067861742052, −12.284941556424297023843276820613, −11.40387938637567169945422522748, −10.41364981986207932061263437977, −9.7423664508356483559433249471, −9.05670290637085286470913801001, −7.9807596476836060374295232684, −7.5945236831305108947355981096, −6.358747671385839411604069102467, −6.02975512591363104469372426840, −5.38457909453007874373913461278, −4.455628714644202462668858121520, −3.88442768311020817280462510644, −3.50307285327289340635869453602, −2.09359757702720964819384848323, −0.80479180177196529756889183281, 0.4147143116572207494787453294, 1.334843155077444740555387759421, 2.462705758133591996552982991298, 3.05275290995014132469836325781, 3.83707168729894827172069527386, 4.59175784222204799718413073494, 5.69040873996603741371668533, 6.16244373181534578240449530089, 6.75555150544111833538399896617, 7.26808140668829578518807963887, 8.53075891959295483426441117012, 9.42828719440897562396296754807, 10.3108949010266839908184482407, 10.8450856150403394809715102806, 11.41457013103095443432374356971, 12.02986755373006023411515447286, 12.58635304453105606228092643533, 13.42503479687244487332020619168, 14.00179724709623472826263330986, 14.48131751128630937671206400978, 15.66641418653908980219270566168, 16.11871632990768814561980463569, 16.56335843368134417688348659770, 17.91698608475611586465795415092, 18.34584994615829991553108044933

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