Properties

Label 1-4033-4033.3528-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.865 - 0.501i$
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 + (0.686 − 0.727i)3-s + (−0.5 + 0.866i)4-s + (0.549 − 0.835i)5-s + (−0.973 − 0.230i)6-s + (−0.0581 + 0.998i)7-s + 8-s + (−0.0581 − 0.998i)9-s + (−0.998 − 0.0581i)10-s + (0.998 − 0.0581i)11-s + (0.286 + 0.957i)12-s + (−0.835 − 0.549i)13-s + (0.893 − 0.448i)14-s + (−0.230 − 0.973i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.686 − 0.727i)3-s + (−0.5 + 0.866i)4-s + (0.549 − 0.835i)5-s + (−0.973 − 0.230i)6-s + (−0.0581 + 0.998i)7-s + 8-s + (−0.0581 − 0.998i)9-s + (−0.998 − 0.0581i)10-s + (0.998 − 0.0581i)11-s + (0.286 + 0.957i)12-s + (−0.835 − 0.549i)13-s + (0.893 − 0.448i)14-s + (−0.230 − 0.973i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.865 - 0.501i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3528, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.865 - 0.501i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4984793104 - 1.852871350i\)
\(L(\frac12)\) \(\approx\) \(0.4984793104 - 1.852871350i\)
\(L(1)\) \(\approx\) \(0.8581533410 - 0.7963150394i\)
\(L(1)\) \(\approx\) \(0.8581533410 - 0.7963150394i\)

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

−18.948422021389449071723582574909, −17.74506944671731012753537798858, −17.29610462628487173396808806089, −16.76367026033552903647606336960, −16.145982306412020533336752090250, −15.09265942618750004536924338074, −14.79337550097702071674352023984, −14.205895943607885799431813485823, −13.69506133730802757403699202380, −13.02270667313716935156733270204, −11.57565896179522683316067117704, −10.85068215983553750516445056124, −10.08469001234754269835262869871, −9.87439737784958280456667111275, −9.03427742470227273695877345146, −8.42130212718905135028957055366, −7.39431897079037976273703723319, −6.97656285432664513352946141395, −6.36466476770553375323473661745, −5.29750405163072875757645101669, −4.54615100787681055380688737743, −3.894522483512861785329759041237, −2.98495767670058268805378784098, −1.99707435557646593619556903222, −1.08718565436367530562566541632, 0.64164777143800199830116938708, 1.44214696130772456552868228302, 2.059369803552501156168245051800, 2.882662489010351613736828037982, 3.460623330224169548622293534476, 4.59303792568019791127715544672, 5.37855709339157287752193714062, 6.22038444029532681738022778828, 7.26910032004483097642680168880, 7.90684440315532126890991753409, 8.6648266431254129601792301383, 9.28862590058858047277385927166, 9.49997859808811471368227519261, 10.42422199147953274283300037138, 11.73196990021080359957852077374, 12.03545222494902940380968711925, 12.57574208015781559485602806627, 13.16633332963875910433339743082, 14.00906196961008827227062089035, 14.49129144986887920471033929536, 15.43676811637295341930468691319, 16.421347979558649744431687650698, 17.08700496260541097222517804259, 17.64827265008604456267273701442, 18.31840552800201040201962977896

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