Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 13 \cdot 31 $
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 − 0.252·3-s + 4-s − 5-s − 0.252·6-s − 1.23·7-s + 8-s − 2.93·9-s − 10-s + 4.08·11-s − 0.252·12-s + 13-s − 1.23·14-s + 0.252·15-s + 16-s − 1.06·17-s − 2.93·18-s − 8.14·19-s − 20-s + 0.313·21-s + 4.08·22-s + 6.59·23-s − 0.252·24-s + 25-s + 26-s + 1.49·27-s − 1.23·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.145·3-s + 0.5·4-s − 0.447·5-s − 0.103·6-s − 0.468·7-s + 0.353·8-s − 0.978·9-s − 0.316·10-s + 1.23·11-s − 0.0729·12-s + 0.277·13-s − 0.331·14-s + 0.0652·15-s + 0.250·16-s − 0.258·17-s − 0.692·18-s − 1.86·19-s − 0.223·20-s + 0.0683·21-s + 0.871·22-s + 1.37·23-s − 0.0515·24-s + 0.200·25-s + 0.196·26-s + 0.288·27-s − 0.234·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4030\)    =    \(2 \cdot 5 \cdot 13 \cdot 31\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4030} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4030,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;13,\;31\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;13,\;31\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 - T \)
31 \( 1 + T \)
good3 \( 1 + 0.252T + 3T^{2} \)
7 \( 1 + 1.23T + 7T^{2} \)
11 \( 1 - 4.08T + 11T^{2} \)
17 \( 1 + 1.06T + 17T^{2} \)
19 \( 1 + 8.14T + 19T^{2} \)
23 \( 1 - 6.59T + 23T^{2} \)
29 \( 1 + 1.85T + 29T^{2} \)
37 \( 1 + 0.537T + 37T^{2} \)
41 \( 1 + 0.693T + 41T^{2} \)
43 \( 1 + 12.5T + 43T^{2} \)
47 \( 1 - 6.79T + 47T^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 - 4.92T + 59T^{2} \)
61 \( 1 - 4.53T + 61T^{2} \)
67 \( 1 + 7.82T + 67T^{2} \)
71 \( 1 - 2.45T + 71T^{2} \)
73 \( 1 + 14.2T + 73T^{2} \)
79 \( 1 - 0.819T + 79T^{2} \)
83 \( 1 + 16.3T + 83T^{2} \)
89 \( 1 - 9.66T + 89T^{2} \)
97 \( 1 + 4.14T + 97T^{2} \)
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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

−8.170078421638837116927505585012, −6.98647458196682498481207981099, −6.56319194967936777080819773370, −5.94662626869916676246187891015, −5.00147649416852869800696144857, −4.21507603821217617716466636578, −3.49869889521612133208211770085, −2.73464016407768138782546903729, −1.53844168845950213150010032283, 0, 1.53844168845950213150010032283, 2.73464016407768138782546903729, 3.49869889521612133208211770085, 4.21507603821217617716466636578, 5.00147649416852869800696144857, 5.94662626869916676246187891015, 6.56319194967936777080819773370, 6.98647458196682498481207981099, 8.170078421638837116927505585012

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