Properties

Label 2-8043-1.1-c1-0-263
Degree $2$
Conductor $8043$
Sign $-1$
Analytic cond. $64.2236$
Root an. cond. $8.01396$
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  − 2.34·2-s − 3-s + 3.51·4-s + 1.78·5-s + 2.34·6-s + 7-s − 3.56·8-s + 9-s − 4.20·10-s + 4.79·11-s − 3.51·12-s − 1.06·13-s − 2.34·14-s − 1.78·15-s + 1.33·16-s − 6.56·17-s − 2.34·18-s + 1.65·19-s + 6.29·20-s − 21-s − 11.2·22-s − 4.89·23-s + 3.56·24-s − 1.79·25-s + 2.50·26-s − 27-s + 3.51·28-s + ⋯
L(s)  = 1  − 1.66·2-s − 0.577·3-s + 1.75·4-s + 0.800·5-s + 0.958·6-s + 0.377·7-s − 1.25·8-s + 0.333·9-s − 1.32·10-s + 1.44·11-s − 1.01·12-s − 0.295·13-s − 0.627·14-s − 0.462·15-s + 0.333·16-s − 1.59·17-s − 0.553·18-s + 0.380·19-s + 1.40·20-s − 0.218·21-s − 2.40·22-s − 1.02·23-s + 0.727·24-s − 0.359·25-s + 0.491·26-s − 0.192·27-s + 0.664·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8043\)    =    \(3 \cdot 7 \cdot 383\)
Sign: $-1$
Analytic conductor: \(64.2236\)
Root analytic conductor: \(8.01396\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8043,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 - T \)
383 \( 1 - T \)
good2 \( 1 + 2.34T + 2T^{2} \)
5 \( 1 - 1.78T + 5T^{2} \)
11 \( 1 - 4.79T + 11T^{2} \)
13 \( 1 + 1.06T + 13T^{2} \)
17 \( 1 + 6.56T + 17T^{2} \)
19 \( 1 - 1.65T + 19T^{2} \)
23 \( 1 + 4.89T + 23T^{2} \)
29 \( 1 - 2.03T + 29T^{2} \)
31 \( 1 + 3.51T + 31T^{2} \)
37 \( 1 - 11.2T + 37T^{2} \)
41 \( 1 - 5.38T + 41T^{2} \)
43 \( 1 + 6.67T + 43T^{2} \)
47 \( 1 - 2.16T + 47T^{2} \)
53 \( 1 + 8.82T + 53T^{2} \)
59 \( 1 + 1.10T + 59T^{2} \)
61 \( 1 + 9.20T + 61T^{2} \)
67 \( 1 + 4.37T + 67T^{2} \)
71 \( 1 - 14.2T + 71T^{2} \)
73 \( 1 - 8.79T + 73T^{2} \)
79 \( 1 + 1.77T + 79T^{2} \)
83 \( 1 + 5.39T + 83T^{2} \)
89 \( 1 + 7.92T + 89T^{2} \)
97 \( 1 + 6.39T + 97T^{2} \)
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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

−7.64716104282857033299113795851, −6.73738742171677035935239510115, −6.42873987080197982337620204977, −5.75929701121831977468528098540, −4.66073929811130486951144455665, −4.00800224041793987460796924957, −2.54433530754281669659859196462, −1.83074942296325924188398453899, −1.18523122697561417811592324449, 0, 1.18523122697561417811592324449, 1.83074942296325924188398453899, 2.54433530754281669659859196462, 4.00800224041793987460796924957, 4.66073929811130486951144455665, 5.75929701121831977468528098540, 6.42873987080197982337620204977, 6.73738742171677035935239510115, 7.64716104282857033299113795851

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