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Properties

Label 1-4024-4024.2011-r0-0-0
Degree $1$
Conductor $4024$
Sign $1$
Analytic cond. $18.6873$
Root an. cond. $18.6873$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 7-s + 9-s + 11-s − 13-s + 15-s − 17-s − 19-s − 21-s − 23-s + 25-s + 27-s + 29-s + 31-s + 33-s − 35-s + 37-s − 39-s − 41-s + 43-s + 45-s − 47-s + 49-s − 51-s + 53-s + 55-s + ⋯
L(s)  = 1  + 3-s + 5-s − 7-s + 9-s + 11-s − 13-s + 15-s − 17-s − 19-s − 21-s − 23-s + 25-s + 27-s + 29-s + 31-s + 33-s − 35-s + 37-s − 39-s − 41-s + 43-s + 45-s − 47-s + 49-s − 51-s + 53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $1$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4024} (2011, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.862545828\)
\(L(\frac12)\) \(\approx\) \(2.862545828\)
\(L(1)\) \(\approx\) \(1.629869768\)
\(L(1)\) \(\approx\) \(1.629869768\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 - T \)
97 \( 1 + T \)
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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

−18.53873991545162153929028209013, −17.79490970605102704555779444367, −17.10085126136038146327500671001, −16.50094617100198497031392775572, −15.61596736533208146612318104676, −15.0137831295747651988972017126, −14.22833041651145985786133269315, −13.829623844830486706890973844, −13.049969199924511218925711541884, −12.57540067498680574368513659576, −11.77833605828621089119725134220, −10.56633318919626946590460196183, −9.908523677614060765971067317141, −9.56969020553727120739970423799, −8.81824795404330512081370825341, −8.24219443695429792599790988700, −7.07337582067727108937731498837, −6.56512098646693250139639628041, −6.0624466267805332266778809186, −4.76371487658436379700880482590, −4.187503408829821939374199224856, −3.277849200125244395885514123125, −2.37458093432355613211497831208, −2.08775446414734509536193868218, −0.85199281253389875376038033836, 0.85199281253389875376038033836, 2.08775446414734509536193868218, 2.37458093432355613211497831208, 3.277849200125244395885514123125, 4.187503408829821939374199224856, 4.76371487658436379700880482590, 6.0624466267805332266778809186, 6.56512098646693250139639628041, 7.07337582067727108937731498837, 8.24219443695429792599790988700, 8.81824795404330512081370825341, 9.56969020553727120739970423799, 9.908523677614060765971067317141, 10.56633318919626946590460196183, 11.77833605828621089119725134220, 12.57540067498680574368513659576, 13.049969199924511218925711541884, 13.829623844830486706890973844, 14.22833041651145985786133269315, 15.0137831295747651988972017126, 15.61596736533208146612318104676, 16.50094617100198497031392775572, 17.10085126136038146327500671001, 17.79490970605102704555779444367, 18.53873991545162153929028209013

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