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Properties

Label 1-4033-4033.2440-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.853 + 0.520i$
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  + (0.5 + 0.866i)2-s + (−0.973 + 0.230i)3-s + (−0.5 + 0.866i)4-s + (0.998 − 0.0581i)5-s + (−0.686 − 0.727i)6-s + (0.893 + 0.448i)7-s − 8-s + (0.893 − 0.448i)9-s + (0.549 + 0.835i)10-s + (−0.549 + 0.835i)11-s + (0.286 − 0.957i)12-s + (0.0581 + 0.998i)13-s + (0.0581 + 0.998i)14-s + (−0.957 + 0.286i)15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.973 + 0.230i)3-s + (−0.5 + 0.866i)4-s + (0.998 − 0.0581i)5-s + (−0.686 − 0.727i)6-s + (0.893 + 0.448i)7-s − 8-s + (0.893 − 0.448i)9-s + (0.549 + 0.835i)10-s + (−0.549 + 0.835i)11-s + (0.286 − 0.957i)12-s + (0.0581 + 0.998i)13-s + (0.0581 + 0.998i)14-s + (−0.957 + 0.286i)15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.853 + 0.520i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.853 + 0.520i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6335816332 + 2.255025874i\)
\(L(\frac12)\) \(\approx\) \(0.6335816332 + 2.255025874i\)
\(L(1)\) \(\approx\) \(0.9725333287 + 0.9466485834i\)
\(L(1)\) \(\approx\) \(0.9725333287 + 0.9466485834i\)

Euler product

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

−18.3670554090870044824844959418, −17.62865839055417979336618791367, −17.11187567770296499727969506566, −16.19339979891247711629999354613, −15.475712868961961661323835505401, −14.38417375111053165046012780320, −13.92778848036451865409208628884, −13.41447449357889270455108541600, −12.60873454369170185041706068031, −12.02925099692253971169366051573, −11.301344351961997284748674859939, −10.58382182767254370077137112845, −10.22927844511011259542833694091, −9.65417388299002954379567691979, −8.347233083883933373747877302895, −7.80145621528542100155242117274, −6.6323346572107540039765444563, −5.80790673492299595230233005401, −5.44946076565013632651893626456, −4.885846830051916705535988014981, −3.91267163579187748000192345977, −2.94663425452452390670188344412, −2.15944287921269243532462822692, −1.12775264523063687641585953818, −0.80651550869473846576926312064, 1.10539361549688127616639127912, 2.01424115965635399295876371741, 2.99440828606489177609509410858, 4.24351177031194707910069174277, 4.79994860587598087153318998012, 5.3467220987176634702682692047, 5.921144704258678818046376396811, 6.65212321998258875709610094289, 7.371385959194451725545010905084, 8.12336703347952476992606379117, 9.09675181843914534577081848783, 9.749009745307680566449802938525, 10.33926184949993820108944634357, 11.44672326568140443341434073219, 12.08936032692409525619641632517, 12.423805004938056215579170846, 13.54886733610191995209391572849, 13.93770528266065631749070270684, 14.74765663611918352712420872078, 15.35808550420941957148677507346, 16.2100742842770759832190311806, 16.54633618621623614644825392899, 17.53168975065827337228133674240, 17.85211603273797322612160807261, 18.17017736757278572826600596032

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