Properties

Label 2-8043-1.1-c1-0-202
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  − 2.41·2-s − 3-s + 3.82·4-s + 1.40·5-s + 2.41·6-s − 7-s − 4.40·8-s + 9-s − 3.38·10-s + 0.776·11-s − 3.82·12-s + 1.29·13-s + 2.41·14-s − 1.40·15-s + 2.97·16-s − 3.64·17-s − 2.41·18-s − 4.65·19-s + 5.35·20-s + 21-s − 1.87·22-s − 7.26·23-s + 4.40·24-s − 3.03·25-s − 3.13·26-s − 27-s − 3.82·28-s + ⋯
L(s)  = 1  − 1.70·2-s − 0.577·3-s + 1.91·4-s + 0.626·5-s + 0.985·6-s − 0.377·7-s − 1.55·8-s + 0.333·9-s − 1.06·10-s + 0.234·11-s − 1.10·12-s + 0.360·13-s + 0.645·14-s − 0.361·15-s + 0.744·16-s − 0.885·17-s − 0.568·18-s − 1.06·19-s + 1.19·20-s + 0.218·21-s − 0.399·22-s − 1.51·23-s + 0.898·24-s − 0.607·25-s − 0.615·26-s − 0.192·27-s − 0.722·28-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 \( 1 + 2.41T + 2T^{2} \)
5 \( 1 - 1.40T + 5T^{2} \)
11 \( 1 - 0.776T + 11T^{2} \)
13 \( 1 - 1.29T + 13T^{2} \)
17 \( 1 + 3.64T + 17T^{2} \)
19 \( 1 + 4.65T + 19T^{2} \)
23 \( 1 + 7.26T + 23T^{2} \)
29 \( 1 - 6.51T + 29T^{2} \)
31 \( 1 - 7.08T + 31T^{2} \)
37 \( 1 + 7.60T + 37T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 - 6.95T + 43T^{2} \)
47 \( 1 - 0.0406T + 47T^{2} \)
53 \( 1 + 10.0T + 53T^{2} \)
59 \( 1 - 7.51T + 59T^{2} \)
61 \( 1 - 4.24T + 61T^{2} \)
67 \( 1 - 8.86T + 67T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 + 2.68T + 79T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 + 0.618T + 89T^{2} \)
97 \( 1 - 3.94T + 97T^{2} \)
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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

−7.61763587910092779718144308560, −6.77962068297129852193146255709, −6.26653172368827668853282698776, −5.93685460531942841286556911153, −4.66839126782527677257593056566, −3.91179715694897482518180531573, −2.52774344337471551077277921545, −2.00677814532342347183209702068, −0.995094610378443863097473111204, 0, 0.995094610378443863097473111204, 2.00677814532342347183209702068, 2.52774344337471551077277921545, 3.91179715694897482518180531573, 4.66839126782527677257593056566, 5.93685460531942841286556911153, 6.26653172368827668853282698776, 6.77962068297129852193146255709, 7.61763587910092779718144308560

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