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 + 1.77·5-s + 6-s − 3.99·7-s − 8-s + 9-s − 1.77·10-s − 0.580·11-s − 12-s + 13-s + 3.99·14-s − 1.77·15-s + 16-s + 6.41·17-s − 18-s − 4.99·19-s + 1.77·20-s + 3.99·21-s + 0.580·22-s − 6.59·23-s + 24-s − 1.84·25-s − 26-s − 27-s − 3.99·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.794·5-s + 0.408·6-s − 1.50·7-s − 0.353·8-s + 0.333·9-s − 0.561·10-s − 0.174·11-s − 0.288·12-s + 0.277·13-s + 1.06·14-s − 0.458·15-s + 0.250·16-s + 1.55·17-s − 0.235·18-s − 1.14·19-s + 0.397·20-s + 0.871·21-s + 0.123·22-s − 1.37·23-s + 0.204·24-s − 0.368·25-s − 0.196·26-s − 0.192·27-s − 0.754·28-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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 + T \)
13 \( 1 - T \)
103 \( 1 - T \)
good5 \( 1 - 1.77T + 5T^{2} \)
7 \( 1 + 3.99T + 7T^{2} \)
11 \( 1 + 0.580T + 11T^{2} \)
17 \( 1 - 6.41T + 17T^{2} \)
19 \( 1 + 4.99T + 19T^{2} \)
23 \( 1 + 6.59T + 23T^{2} \)
29 \( 1 - 1.31T + 29T^{2} \)
31 \( 1 - 6.05T + 31T^{2} \)
37 \( 1 - 6.39T + 37T^{2} \)
41 \( 1 - 6.04T + 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 + 6.62T + 47T^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 + 2.66T + 67T^{2} \)
71 \( 1 - 3.84T + 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 - 2.69T + 79T^{2} \)
83 \( 1 - 7.90T + 83T^{2} \)
89 \( 1 + 3.95T + 89T^{2} \)
97 \( 1 - 8.16T + 97T^{2} \)
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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

−7.71830054067671860479979804770, −6.48792261159569398382976250081, −6.05552756325167038200323892037, −5.99713332775557076293989844627, −4.74044710666842190976505124647, −3.78255092591082478791654089450, −2.95321217729174614152826210791, −2.13309196936108115146503501238, −1.05626427956969767782349388641, 0, 1.05626427956969767782349388641, 2.13309196936108115146503501238, 2.95321217729174614152826210791, 3.78255092591082478791654089450, 4.74044710666842190976505124647, 5.99713332775557076293989844627, 6.05552756325167038200323892037, 6.48792261159569398382976250081, 7.71830054067671860479979804770

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