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 − 0.453·5-s − 6-s + 3.87·7-s − 8-s + 9-s + 0.453·10-s − 1.29·11-s + 12-s − 13-s − 3.87·14-s − 0.453·15-s + 16-s − 0.204·17-s − 18-s − 0.475·19-s − 0.453·20-s + 3.87·21-s + 1.29·22-s + 1.84·23-s − 24-s − 4.79·25-s + 26-s + 27-s + 3.87·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.202·5-s − 0.408·6-s + 1.46·7-s − 0.353·8-s + 0.333·9-s + 0.143·10-s − 0.389·11-s + 0.288·12-s − 0.277·13-s − 1.03·14-s − 0.117·15-s + 0.250·16-s − 0.0494·17-s − 0.235·18-s − 0.109·19-s − 0.101·20-s + 0.846·21-s + 0.275·22-s + 0.385·23-s − 0.204·24-s − 0.958·25-s + 0.196·26-s + 0.192·27-s + 0.732·28-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 \( 1 + 0.453T + 5T^{2} \)
7 \( 1 - 3.87T + 7T^{2} \)
11 \( 1 + 1.29T + 11T^{2} \)
17 \( 1 + 0.204T + 17T^{2} \)
19 \( 1 + 0.475T + 19T^{2} \)
23 \( 1 - 1.84T + 23T^{2} \)
29 \( 1 + 5.66T + 29T^{2} \)
31 \( 1 + 7.72T + 31T^{2} \)
37 \( 1 - 9.68T + 37T^{2} \)
41 \( 1 + 7.38T + 41T^{2} \)
43 \( 1 + 4.47T + 43T^{2} \)
47 \( 1 + 3.11T + 47T^{2} \)
53 \( 1 + 2.30T + 53T^{2} \)
59 \( 1 + 13.6T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 - 0.678T + 67T^{2} \)
71 \( 1 - 5.87T + 71T^{2} \)
73 \( 1 + 3.56T + 73T^{2} \)
79 \( 1 - 0.0379T + 79T^{2} \)
83 \( 1 + 7.81T + 83T^{2} \)
89 \( 1 - 8.08T + 89T^{2} \)
97 \( 1 + 2.30T + 97T^{2} \)
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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

−7.79671118859045016015539065569, −7.17957356153019319365577347667, −6.19241277065225228640998668046, −5.33605452150117573633874707473, −4.68158310312501095116618109390, −3.84456083495646570633400579273, −2.93390962901739565942721000094, −1.96983854029079652288219520623, −1.48464769958654406166094910247, 0, 1.48464769958654406166094910247, 1.96983854029079652288219520623, 2.93390962901739565942721000094, 3.84456083495646570633400579273, 4.68158310312501095116618109390, 5.33605452150117573633874707473, 6.19241277065225228640998668046, 7.17957356153019319365577347667, 7.79671118859045016015539065569

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