Properties

Label 2-4023-1.1-c1-0-62
Degree $2$
Conductor $4023$
Sign $-1$
Analytic cond. $32.1238$
Root an. cond. $5.66778$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.45·2-s + 4.00·4-s − 3.55·5-s − 0.769·7-s − 4.91·8-s + 8.71·10-s − 5.40·11-s − 2.95·13-s + 1.88·14-s + 4.03·16-s + 8.13·17-s − 2.72·19-s − 14.2·20-s + 13.2·22-s − 6.87·23-s + 7.63·25-s + 7.23·26-s − 3.08·28-s + 1.15·29-s + 10.1·31-s − 0.0518·32-s − 19.9·34-s + 2.73·35-s + 8.27·37-s + 6.66·38-s + 17.4·40-s − 3.54·41-s + ⋯
L(s)  = 1  − 1.73·2-s + 2.00·4-s − 1.58·5-s − 0.290·7-s − 1.73·8-s + 2.75·10-s − 1.62·11-s − 0.818·13-s + 0.503·14-s + 1.00·16-s + 1.97·17-s − 0.624·19-s − 3.18·20-s + 2.82·22-s − 1.43·23-s + 1.52·25-s + 1.41·26-s − 0.582·28-s + 0.214·29-s + 1.81·31-s − 0.00916·32-s − 3.41·34-s + 0.462·35-s + 1.35·37-s + 1.08·38-s + 2.76·40-s − 0.552·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4023\)    =    \(3^{3} \cdot 149\)
Sign: $-1$
Analytic conductor: \(32.1238\)
Root analytic conductor: \(5.66778\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4023,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
149 \( 1 + T \)
good2 \( 1 + 2.45T + 2T^{2} \)
5 \( 1 + 3.55T + 5T^{2} \)
7 \( 1 + 0.769T + 7T^{2} \)
11 \( 1 + 5.40T + 11T^{2} \)
13 \( 1 + 2.95T + 13T^{2} \)
17 \( 1 - 8.13T + 17T^{2} \)
19 \( 1 + 2.72T + 19T^{2} \)
23 \( 1 + 6.87T + 23T^{2} \)
29 \( 1 - 1.15T + 29T^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
37 \( 1 - 8.27T + 37T^{2} \)
41 \( 1 + 3.54T + 41T^{2} \)
43 \( 1 - 4.30T + 43T^{2} \)
47 \( 1 + 5.42T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 - 1.20T + 59T^{2} \)
61 \( 1 - 8.45T + 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 - 1.70T + 71T^{2} \)
73 \( 1 + 15.0T + 73T^{2} \)
79 \( 1 + 6.79T + 79T^{2} \)
83 \( 1 - 9.19T + 83T^{2} \)
89 \( 1 - 5.62T + 89T^{2} \)
97 \( 1 + 1.92T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.036992969951948604468422497958, −7.78521068643592797232827224492, −7.10830836788694609272710033283, −6.17494858673151736770314706864, −5.13241105526716674139280438927, −4.13866433804854409898119651324, −3.05272133372443925085471644145, −2.38756115206285130549927546290, −0.844629369497922400354295152258, 0, 0.844629369497922400354295152258, 2.38756115206285130549927546290, 3.05272133372443925085471644145, 4.13866433804854409898119651324, 5.13241105526716674139280438927, 6.17494858673151736770314706864, 7.10830836788694609272710033283, 7.78521068643592797232827224492, 8.036992969951948604468422497958

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