Properties

Degree 2
Conductor $ 29 \cdot 139 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.13·2-s + 1.15·3-s − 0.712·4-s + 3.10·5-s − 1.31·6-s − 1.02·7-s + 3.07·8-s − 1.66·9-s − 3.52·10-s − 1.83·11-s − 0.823·12-s + 3.48·13-s + 1.16·14-s + 3.59·15-s − 2.06·16-s − 1.09·17-s + 1.88·18-s − 2.20·19-s − 2.21·20-s − 1.18·21-s + 2.08·22-s − 1.23·23-s + 3.55·24-s + 4.66·25-s − 3.95·26-s − 5.39·27-s + 0.728·28-s + ⋯
L(s)  = 1  − 0.802·2-s + 0.667·3-s − 0.356·4-s + 1.39·5-s − 0.535·6-s − 0.386·7-s + 1.08·8-s − 0.554·9-s − 1.11·10-s − 0.553·11-s − 0.237·12-s + 0.966·13-s + 0.310·14-s + 0.928·15-s − 0.516·16-s − 0.265·17-s + 0.444·18-s − 0.506·19-s − 0.495·20-s − 0.258·21-s + 0.444·22-s − 0.257·23-s + 0.726·24-s + 0.933·25-s − 0.775·26-s − 1.03·27-s + 0.137·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4031\)    =    \(29 \cdot 139\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4031} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4031,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{29,\;139\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{29,\;139\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad29 \( 1 + T \)
139 \( 1 + T \)
good2 \( 1 + 1.13T + 2T^{2} \)
3 \( 1 - 1.15T + 3T^{2} \)
5 \( 1 - 3.10T + 5T^{2} \)
7 \( 1 + 1.02T + 7T^{2} \)
11 \( 1 + 1.83T + 11T^{2} \)
13 \( 1 - 3.48T + 13T^{2} \)
17 \( 1 + 1.09T + 17T^{2} \)
19 \( 1 + 2.20T + 19T^{2} \)
23 \( 1 + 1.23T + 23T^{2} \)
31 \( 1 + 4.93T + 31T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
41 \( 1 + 3.31T + 41T^{2} \)
43 \( 1 - 7.70T + 43T^{2} \)
47 \( 1 - 7.43T + 47T^{2} \)
53 \( 1 + 2.52T + 53T^{2} \)
59 \( 1 - 4.84T + 59T^{2} \)
61 \( 1 + 5.33T + 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 + 8.54T + 71T^{2} \)
73 \( 1 - 6.94T + 73T^{2} \)
79 \( 1 - 6.59T + 79T^{2} \)
83 \( 1 + 9.84T + 83T^{2} \)
89 \( 1 + 8.43T + 89T^{2} \)
97 \( 1 + 1.62T + 97T^{2} \)
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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

−8.352824355640580660856094773939, −7.59614542436745861939731038967, −6.66974198048308357526374257764, −5.81023632042254861649698458360, −5.31592564116767649922303559248, −4.15168356629724879124209771218, −3.23473056013717411716459103448, −2.23663849958907735705905354558, −1.52220871472409761844367247380, 0, 1.52220871472409761844367247380, 2.23663849958907735705905354558, 3.23473056013717411716459103448, 4.15168356629724879124209771218, 5.31592564116767649922303559248, 5.81023632042254861649698458360, 6.66974198048308357526374257764, 7.59614542436745861939731038967, 8.352824355640580660856094773939

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