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 − 3.83·5-s − 6-s − 3.29·7-s − 8-s + 9-s + 3.83·10-s − 1.08·11-s + 12-s + 13-s + 3.29·14-s − 3.83·15-s + 16-s − 5.28·17-s − 18-s + 7.52·19-s − 3.83·20-s − 3.29·21-s + 1.08·22-s − 4.86·23-s − 24-s + 9.69·25-s − 26-s + 27-s − 3.29·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.71·5-s − 0.408·6-s − 1.24·7-s − 0.353·8-s + 0.333·9-s + 1.21·10-s − 0.327·11-s + 0.288·12-s + 0.277·13-s + 0.881·14-s − 0.989·15-s + 0.250·16-s − 1.28·17-s − 0.235·18-s + 1.72·19-s − 0.857·20-s − 0.719·21-s + 0.231·22-s − 1.01·23-s − 0.204·24-s + 1.93·25-s − 0.196·26-s + 0.192·27-s − 0.623·28-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 \( 1 + 3.83T + 5T^{2} \)
7 \( 1 + 3.29T + 7T^{2} \)
11 \( 1 + 1.08T + 11T^{2} \)
17 \( 1 + 5.28T + 17T^{2} \)
19 \( 1 - 7.52T + 19T^{2} \)
23 \( 1 + 4.86T + 23T^{2} \)
29 \( 1 + 8.02T + 29T^{2} \)
31 \( 1 - 3.95T + 31T^{2} \)
37 \( 1 - 3.95T + 37T^{2} \)
41 \( 1 - 5.35T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 - 1.82T + 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 + 14.8T + 61T^{2} \)
67 \( 1 - 9.36T + 67T^{2} \)
71 \( 1 + 9.48T + 71T^{2} \)
73 \( 1 + 5.79T + 73T^{2} \)
79 \( 1 - 6.22T + 79T^{2} \)
83 \( 1 - 2.49T + 83T^{2} \)
89 \( 1 - 5.79T + 89T^{2} \)
97 \( 1 + 5.65T + 97T^{2} \)
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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

−7.62244123911284425671986330538, −7.14751360193525594299418396943, −6.35229092588951707371710619383, −5.52937279518425523203959805676, −4.22517435963387747353911448136, −3.87952746273639900958356060958, −3.04208494184992178749068768260, −2.43318935538383437149290873078, −0.915475189864899244638395204044, 0, 0.915475189864899244638395204044, 2.43318935538383437149290873078, 3.04208494184992178749068768260, 3.87952746273639900958356060958, 4.22517435963387747353911448136, 5.52937279518425523203959805676, 6.35229092588951707371710619383, 7.14751360193525594299418396943, 7.62244123911284425671986330538

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