Properties

Degree $2$
Conductor $8034$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 2.22·5-s − 6-s + 1.37·7-s − 8-s + 9-s + 2.22·10-s + 3.97·11-s + 12-s − 13-s − 1.37·14-s − 2.22·15-s + 16-s + 2.05·17-s − 18-s − 6.55·19-s − 2.22·20-s + 1.37·21-s − 3.97·22-s − 0.577·23-s − 24-s − 0.0525·25-s + 26-s + 27-s + 1.37·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.994·5-s − 0.408·6-s + 0.518·7-s − 0.353·8-s + 0.333·9-s + 0.703·10-s + 1.19·11-s + 0.288·12-s − 0.277·13-s − 0.366·14-s − 0.574·15-s + 0.250·16-s + 0.499·17-s − 0.235·18-s − 1.50·19-s − 0.497·20-s + 0.299·21-s − 0.846·22-s − 0.120·23-s − 0.204·24-s − 0.0105·25-s + 0.196·26-s + 0.192·27-s + 0.259·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8034\)    =    \(2 \cdot 3 \cdot 13 \cdot 103\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{8034} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8034,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
3 \( 1 - T \)
13 \( 1 + T \)
103 \( 1 - T \)
good5 \( 1 + 2.22T + 5T^{2} \)
7 \( 1 - 1.37T + 7T^{2} \)
11 \( 1 - 3.97T + 11T^{2} \)
17 \( 1 - 2.05T + 17T^{2} \)
19 \( 1 + 6.55T + 19T^{2} \)
23 \( 1 + 0.577T + 23T^{2} \)
29 \( 1 + 3.79T + 29T^{2} \)
31 \( 1 - 2.84T + 31T^{2} \)
37 \( 1 + 3.41T + 37T^{2} \)
41 \( 1 + 6.35T + 41T^{2} \)
43 \( 1 - 12.0T + 43T^{2} \)
47 \( 1 - 2.35T + 47T^{2} \)
53 \( 1 + 9.83T + 53T^{2} \)
59 \( 1 - 5.32T + 59T^{2} \)
61 \( 1 - 4.72T + 61T^{2} \)
67 \( 1 - 2.48T + 67T^{2} \)
71 \( 1 + 9.14T + 71T^{2} \)
73 \( 1 - 9.71T + 73T^{2} \)
79 \( 1 + 1.40T + 79T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 + 17.1T + 89T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−7.65192221473061857629370668741, −6.99939643516900016914624587087, −6.40085959114725607214361210566, −5.43759840836449414696592236884, −4.28770160025885011767276732010, −3.99770824218729080193467872001, −3.06495735488709035301651697358, −2.06902688469757288171233503230, −1.27756694770733209873428175938, 0, 1.27756694770733209873428175938, 2.06902688469757288171233503230, 3.06495735488709035301651697358, 3.99770824218729080193467872001, 4.28770160025885011767276732010, 5.43759840836449414696592236884, 6.40085959114725607214361210566, 6.99939643516900016914624587087, 7.65192221473061857629370668741

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