Properties

Label 2-8043-1.1-c1-0-234
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  + 1.33·2-s − 3-s − 0.213·4-s − 2.23·5-s − 1.33·6-s + 7-s − 2.95·8-s + 9-s − 2.99·10-s + 1.45·11-s + 0.213·12-s − 2.95·13-s + 1.33·14-s + 2.23·15-s − 3.52·16-s + 2.07·17-s + 1.33·18-s + 1.00·19-s + 0.477·20-s − 21-s + 1.94·22-s + 0.269·23-s + 2.95·24-s + 0.00438·25-s − 3.94·26-s − 27-s − 0.213·28-s + ⋯
L(s)  = 1  + 0.945·2-s − 0.577·3-s − 0.106·4-s − 1.00·5-s − 0.545·6-s + 0.377·7-s − 1.04·8-s + 0.333·9-s − 0.945·10-s + 0.437·11-s + 0.0616·12-s − 0.818·13-s + 0.357·14-s + 0.577·15-s − 0.881·16-s + 0.504·17-s + 0.315·18-s + 0.230·19-s + 0.106·20-s − 0.218·21-s + 0.413·22-s + 0.0561·23-s + 0.603·24-s + 0.000876·25-s − 0.773·26-s − 0.192·27-s − 0.0403·28-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 \( 1 - 1.33T + 2T^{2} \)
5 \( 1 + 2.23T + 5T^{2} \)
11 \( 1 - 1.45T + 11T^{2} \)
13 \( 1 + 2.95T + 13T^{2} \)
17 \( 1 - 2.07T + 17T^{2} \)
19 \( 1 - 1.00T + 19T^{2} \)
23 \( 1 - 0.269T + 23T^{2} \)
29 \( 1 - 3.54T + 29T^{2} \)
31 \( 1 - 9.24T + 31T^{2} \)
37 \( 1 + 1.52T + 37T^{2} \)
41 \( 1 - 0.864T + 41T^{2} \)
43 \( 1 + 8.66T + 43T^{2} \)
47 \( 1 - 6.60T + 47T^{2} \)
53 \( 1 + 2.69T + 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 - 0.682T + 61T^{2} \)
67 \( 1 + 0.953T + 67T^{2} \)
71 \( 1 - 5.11T + 71T^{2} \)
73 \( 1 + 3.39T + 73T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 + 6.82T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 - 3.72T + 97T^{2} \)
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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

−7.45759497322909549951679756895, −6.57685634404268778469880798030, −6.05620721339016033102345979624, −5.02194102048596707552070117250, −4.81751636420939977646123094889, −4.03114502697181661429045213743, −3.40361714343582264141936031575, −2.51360634720526538288297811572, −1.09236853352556820434116110608, 0, 1.09236853352556820434116110608, 2.51360634720526538288297811572, 3.40361714343582264141936031575, 4.03114502697181661429045213743, 4.81751636420939977646123094889, 5.02194102048596707552070117250, 6.05620721339016033102345979624, 6.57685634404268778469880798030, 7.45759497322909549951679756895

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