Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 13 \cdot 103 $
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 − 3.54·5-s − 6-s + 1.31·7-s − 8-s + 9-s + 3.54·10-s + 3.27·11-s + 12-s + 13-s − 1.31·14-s − 3.54·15-s + 16-s + 2.46·17-s − 18-s + 1.21·19-s − 3.54·20-s + 1.31·21-s − 3.27·22-s − 4.22·23-s − 24-s + 7.59·25-s − 26-s + 27-s + 1.31·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.58·5-s − 0.408·6-s + 0.496·7-s − 0.353·8-s + 0.333·9-s + 1.12·10-s + 0.987·11-s + 0.288·12-s + 0.277·13-s − 0.350·14-s − 0.916·15-s + 0.250·16-s + 0.597·17-s − 0.235·18-s + 0.279·19-s − 0.793·20-s + 0.286·21-s − 0.698·22-s − 0.879·23-s − 0.204·24-s + 1.51·25-s − 0.196·26-s + 0.192·27-s + 0.248·28-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

\( d \)  =  \(2\)
\( N \)  =  \(8034\)    =    \(2 \cdot 3 \cdot 13 \cdot 103\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8034} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 8034,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;13,\;103\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;13,\;103\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
13 \( 1 - T \)
103 \( 1 + T \)
good5 \( 1 + 3.54T + 5T^{2} \)
7 \( 1 - 1.31T + 7T^{2} \)
11 \( 1 - 3.27T + 11T^{2} \)
17 \( 1 - 2.46T + 17T^{2} \)
19 \( 1 - 1.21T + 19T^{2} \)
23 \( 1 + 4.22T + 23T^{2} \)
29 \( 1 + 7.93T + 29T^{2} \)
31 \( 1 + 3.85T + 31T^{2} \)
37 \( 1 + 5.26T + 37T^{2} \)
41 \( 1 - 9.66T + 41T^{2} \)
43 \( 1 + 6.54T + 43T^{2} \)
47 \( 1 + 8.05T + 47T^{2} \)
53 \( 1 - 0.0283T + 53T^{2} \)
59 \( 1 + 11.0T + 59T^{2} \)
61 \( 1 - 2.25T + 61T^{2} \)
67 \( 1 - 14.9T + 67T^{2} \)
71 \( 1 + 0.0351T + 71T^{2} \)
73 \( 1 - 7.18T + 73T^{2} \)
79 \( 1 - 4.16T + 79T^{2} \)
83 \( 1 + 8.86T + 83T^{2} \)
89 \( 1 + 7.73T + 89T^{2} \)
97 \( 1 - 9.04T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.76116222987683275457750569748, −7.10866974295657809886244469758, −6.38534766998278485162597184919, −5.37534870821522134478876844506, −4.42375132163757755849075764814, −3.65246169715446293534193073571, −3.37020701756405924477592348853, −2.02802309764199751788894174472, −1.21670443082506356069760358187, 0, 1.21670443082506356069760358187, 2.02802309764199751788894174472, 3.37020701756405924477592348853, 3.65246169715446293534193073571, 4.42375132163757755849075764814, 5.37534870821522134478876844506, 6.38534766998278485162597184919, 7.10866974295657809886244469758, 7.76116222987683275457750569748

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