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 − 3.81·5-s − 6-s + 4.94·7-s − 8-s + 9-s + 3.81·10-s + 2.65·11-s + 12-s − 13-s − 4.94·14-s − 3.81·15-s + 16-s − 4.45·17-s − 18-s + 3.06·19-s − 3.81·20-s + 4.94·21-s − 2.65·22-s − 3.71·23-s − 24-s + 9.55·25-s + 26-s + 27-s + 4.94·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.70·5-s − 0.408·6-s + 1.86·7-s − 0.353·8-s + 0.333·9-s + 1.20·10-s + 0.801·11-s + 0.288·12-s − 0.277·13-s − 1.32·14-s − 0.985·15-s + 0.250·16-s − 1.08·17-s − 0.235·18-s + 0.703·19-s − 0.853·20-s + 1.07·21-s − 0.566·22-s − 0.774·23-s − 0.204·24-s + 1.91·25-s + 0.196·26-s + 0.192·27-s + 0.933·28-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 \( 1 + 3.81T + 5T^{2} \)
7 \( 1 - 4.94T + 7T^{2} \)
11 \( 1 - 2.65T + 11T^{2} \)
17 \( 1 + 4.45T + 17T^{2} \)
19 \( 1 - 3.06T + 19T^{2} \)
23 \( 1 + 3.71T + 23T^{2} \)
29 \( 1 - 5.31T + 29T^{2} \)
31 \( 1 + 4.66T + 31T^{2} \)
37 \( 1 + 5.87T + 37T^{2} \)
41 \( 1 + 6.56T + 41T^{2} \)
43 \( 1 + 5.51T + 43T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 + 4.64T + 61T^{2} \)
67 \( 1 + 1.96T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 + 1.93T + 73T^{2} \)
79 \( 1 + 5.22T + 79T^{2} \)
83 \( 1 - 3.46T + 83T^{2} \)
89 \( 1 - 8.23T + 89T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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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

−7.64100731885909126478859822116, −7.18299858025305403991128678783, −6.46585591333147962095014743485, −5.07947784205571483372460350148, −4.60293086852776770472583603670, −3.88086440639848355905792049828, −3.14878151458651413357423153956, −1.96597244260061186175447954534, −1.31747417707460182452185714681, 0, 1.31747417707460182452185714681, 1.96597244260061186175447954534, 3.14878151458651413357423153956, 3.88086440639848355905792049828, 4.60293086852776770472583603670, 5.07947784205571483372460350148, 6.46585591333147962095014743485, 7.18299858025305403991128678783, 7.64100731885909126478859822116

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