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 + 3.07·5-s − 6-s − 1.20·7-s − 8-s + 9-s − 3.07·10-s + 1.64·11-s + 12-s − 13-s + 1.20·14-s + 3.07·15-s + 16-s + 5.16·17-s − 18-s − 6.97·19-s + 3.07·20-s − 1.20·21-s − 1.64·22-s − 4.45·23-s − 24-s + 4.42·25-s + 26-s + 27-s − 1.20·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.37·5-s − 0.408·6-s − 0.454·7-s − 0.353·8-s + 0.333·9-s − 0.970·10-s + 0.495·11-s + 0.288·12-s − 0.277·13-s + 0.321·14-s + 0.792·15-s + 0.250·16-s + 1.25·17-s − 0.235·18-s − 1.59·19-s + 0.686·20-s − 0.262·21-s − 0.350·22-s − 0.928·23-s − 0.204·24-s + 0.885·25-s + 0.196·26-s + 0.192·27-s − 0.227·28-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 \( 1 - 3.07T + 5T^{2} \)
7 \( 1 + 1.20T + 7T^{2} \)
11 \( 1 - 1.64T + 11T^{2} \)
17 \( 1 - 5.16T + 17T^{2} \)
19 \( 1 + 6.97T + 19T^{2} \)
23 \( 1 + 4.45T + 23T^{2} \)
29 \( 1 + 5.46T + 29T^{2} \)
31 \( 1 + 0.105T + 31T^{2} \)
37 \( 1 + 4.85T + 37T^{2} \)
41 \( 1 + 1.62T + 41T^{2} \)
43 \( 1 + 12.7T + 43T^{2} \)
47 \( 1 + 9.64T + 47T^{2} \)
53 \( 1 + 8.08T + 53T^{2} \)
59 \( 1 + 4.90T + 59T^{2} \)
61 \( 1 + 2.19T + 61T^{2} \)
67 \( 1 + 15.1T + 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 - 0.0855T + 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 - 5.79T + 83T^{2} \)
89 \( 1 - 4.12T + 89T^{2} \)
97 \( 1 + 3.96T + 97T^{2} \)
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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

−7.64519394056587568895200002042, −6.66054606803140029720774184544, −6.33674976426227144088012389643, −5.61240652750012187399024408398, −4.72837202351113348525910983734, −3.61019049402305205129348588271, −2.98541251421955017385538127518, −1.79041168928335264322328387509, −1.73449971740347984951326747604, 0, 1.73449971740347984951326747604, 1.79041168928335264322328387509, 2.98541251421955017385538127518, 3.61019049402305205129348588271, 4.72837202351113348525910983734, 5.61240652750012187399024408398, 6.33674976426227144088012389643, 6.66054606803140029720774184544, 7.64519394056587568895200002042

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