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 − 1.56·3-s + 4-s − 5-s − 1.56·6-s − 1.49·7-s + 8-s − 0.540·9-s − 10-s − 1.02·11-s − 1.56·12-s + 13-s − 1.49·14-s + 1.56·15-s + 16-s + 1.43·17-s − 0.540·18-s + 2.94·19-s − 20-s + 2.33·21-s − 1.02·22-s + 1.14·23-s − 1.56·24-s + 25-s + 26-s + 5.55·27-s − 1.49·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.905·3-s + 0.5·4-s − 0.447·5-s − 0.640·6-s − 0.563·7-s + 0.353·8-s − 0.180·9-s − 0.316·10-s − 0.309·11-s − 0.452·12-s + 0.277·13-s − 0.398·14-s + 0.404·15-s + 0.250·16-s + 0.347·17-s − 0.127·18-s + 0.675·19-s − 0.223·20-s + 0.510·21-s − 0.218·22-s + 0.237·23-s − 0.320·24-s + 0.200·25-s + 0.196·26-s + 1.06·27-s − 0.281·28-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 \( 1 + 1.56T + 3T^{2} \)
7 \( 1 + 1.49T + 7T^{2} \)
11 \( 1 + 1.02T + 11T^{2} \)
17 \( 1 - 1.43T + 17T^{2} \)
19 \( 1 - 2.94T + 19T^{2} \)
23 \( 1 - 1.14T + 23T^{2} \)
29 \( 1 - 6.02T + 29T^{2} \)
37 \( 1 + 6.22T + 37T^{2} \)
41 \( 1 - 3.45T + 41T^{2} \)
43 \( 1 - 4.32T + 43T^{2} \)
47 \( 1 + 4.37T + 47T^{2} \)
53 \( 1 + 9.04T + 53T^{2} \)
59 \( 1 + 8.83T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 9.32T + 67T^{2} \)
71 \( 1 + 15.9T + 71T^{2} \)
73 \( 1 + 4.75T + 73T^{2} \)
79 \( 1 + 5.97T + 79T^{2} \)
83 \( 1 - 0.949T + 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−7.898242782008704109030244986810, −7.15854777946553951581904272649, −6.37570296935615149417514420209, −5.86999206013247433929774848894, −5.07357707870403998563344845520, −4.46978052464605268275228356593, −3.34345103794946846890114531416, −2.85624532151780667063459694251, −1.30999270594204474368782989440, 0, 1.30999270594204474368782989440, 2.85624532151780667063459694251, 3.34345103794946846890114531416, 4.46978052464605268275228356593, 5.07357707870403998563344845520, 5.86999206013247433929774848894, 6.37570296935615149417514420209, 7.15854777946553951581904272649, 7.898242782008704109030244986810

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