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 + 2.43·3-s + 4-s − 5-s + 2.43·6-s − 3.35·7-s + 8-s + 2.93·9-s − 10-s − 3.03·11-s + 2.43·12-s + 13-s − 3.35·14-s − 2.43·15-s + 16-s − 4.12·17-s + 2.93·18-s − 2.24·19-s − 20-s − 8.18·21-s − 3.03·22-s − 2.50·23-s + 2.43·24-s + 25-s + 26-s − 0.158·27-s − 3.35·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.40·3-s + 0.5·4-s − 0.447·5-s + 0.994·6-s − 1.26·7-s + 0.353·8-s + 0.978·9-s − 0.316·10-s − 0.915·11-s + 0.703·12-s + 0.277·13-s − 0.897·14-s − 0.629·15-s + 0.250·16-s − 0.999·17-s + 0.691·18-s − 0.515·19-s − 0.223·20-s − 1.78·21-s − 0.647·22-s − 0.522·23-s + 0.497·24-s + 0.200·25-s + 0.196·26-s − 0.0304·27-s − 0.634·28-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 \( 1 - 2.43T + 3T^{2} \)
7 \( 1 + 3.35T + 7T^{2} \)
11 \( 1 + 3.03T + 11T^{2} \)
17 \( 1 + 4.12T + 17T^{2} \)
19 \( 1 + 2.24T + 19T^{2} \)
23 \( 1 + 2.50T + 23T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
37 \( 1 - 8.42T + 37T^{2} \)
41 \( 1 - 3.29T + 41T^{2} \)
43 \( 1 + 2.39T + 43T^{2} \)
47 \( 1 + 5.72T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 - 2.41T + 59T^{2} \)
61 \( 1 + 1.65T + 61T^{2} \)
67 \( 1 - 10.9T + 67T^{2} \)
71 \( 1 + 6.06T + 71T^{2} \)
73 \( 1 + 1.74T + 73T^{2} \)
79 \( 1 - 9.95T + 79T^{2} \)
83 \( 1 - 0.183T + 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 - 5.54T + 97T^{2} \)
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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

−7.914362136331158545876156548787, −7.53654996332636203658157455226, −6.56317914479734711214185186483, −5.98364943828795711269276144143, −4.86317830940873443169435975126, −3.92725483185964929202318235836, −3.45864112595461403444891569306, −2.67187855685231552205747493125, −2.00188938433334472994055470085, 0, 2.00188938433334472994055470085, 2.67187855685231552205747493125, 3.45864112595461403444891569306, 3.92725483185964929202318235836, 4.86317830940873443169435975126, 5.98364943828795711269276144143, 6.56317914479734711214185186483, 7.53654996332636203658157455226, 7.914362136331158545876156548787

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