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 − 2.94·5-s + 6-s − 3.14·7-s − 8-s + 9-s + 2.94·10-s + 0.137·11-s − 12-s + 13-s + 3.14·14-s + 2.94·15-s + 16-s − 2.74·17-s − 18-s − 4.14·19-s − 2.94·20-s + 3.14·21-s − 0.137·22-s + 7.19·23-s + 24-s + 3.65·25-s − 26-s − 27-s − 3.14·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.31·5-s + 0.408·6-s − 1.19·7-s − 0.353·8-s + 0.333·9-s + 0.930·10-s + 0.0413·11-s − 0.288·12-s + 0.277·13-s + 0.841·14-s + 0.759·15-s + 0.250·16-s − 0.666·17-s − 0.235·18-s − 0.951·19-s − 0.657·20-s + 0.687·21-s − 0.0292·22-s + 1.49·23-s + 0.204·24-s + 0.730·25-s − 0.196·26-s − 0.192·27-s − 0.595·28-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 \( 1 + 2.94T + 5T^{2} \)
7 \( 1 + 3.14T + 7T^{2} \)
11 \( 1 - 0.137T + 11T^{2} \)
17 \( 1 + 2.74T + 17T^{2} \)
19 \( 1 + 4.14T + 19T^{2} \)
23 \( 1 - 7.19T + 23T^{2} \)
29 \( 1 + 10.1T + 29T^{2} \)
31 \( 1 - 4.60T + 31T^{2} \)
37 \( 1 + 0.829T + 37T^{2} \)
41 \( 1 - 4.33T + 41T^{2} \)
43 \( 1 + 9.96T + 43T^{2} \)
47 \( 1 - 6.77T + 47T^{2} \)
53 \( 1 - 5.15T + 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 + 8.47T + 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 + 4.30T + 71T^{2} \)
73 \( 1 - 1.09T + 73T^{2} \)
79 \( 1 + 2.76T + 79T^{2} \)
83 \( 1 - 8.49T + 83T^{2} \)
89 \( 1 - 3.36T + 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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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

−7.37111788504496307592901713120, −6.89976081891060835415015005620, −6.36695021056266406044981324674, −5.54403105756586853682163993291, −4.56005903007007039348089732863, −3.80768723178610052054885029856, −3.20797307415258400429063837591, −2.14840037383694094793760962160, −0.78019607539215726766331184144, 0, 0.78019607539215726766331184144, 2.14840037383694094793760962160, 3.20797307415258400429063837591, 3.80768723178610052054885029856, 4.56005903007007039348089732863, 5.54403105756586853682163993291, 6.36695021056266406044981324674, 6.89976081891060835415015005620, 7.37111788504496307592901713120

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