Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 13 \cdot 31 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2.04·3-s + 4-s − 5-s − 2.04·6-s + 3.38·7-s + 8-s + 1.19·9-s − 10-s + 4.04·11-s − 2.04·12-s − 13-s + 3.38·14-s + 2.04·15-s + 16-s − 0.266·17-s + 1.19·18-s + 0.644·19-s − 20-s − 6.92·21-s + 4.04·22-s − 2.41·23-s − 2.04·24-s + 25-s − 26-s + 3.69·27-s + 3.38·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.18·3-s + 0.5·4-s − 0.447·5-s − 0.836·6-s + 1.27·7-s + 0.353·8-s + 0.398·9-s − 0.316·10-s + 1.21·11-s − 0.591·12-s − 0.277·13-s + 0.903·14-s + 0.528·15-s + 0.250·16-s − 0.0645·17-s + 0.281·18-s + 0.147·19-s − 0.223·20-s − 1.51·21-s + 0.862·22-s − 0.502·23-s − 0.418·24-s + 0.200·25-s − 0.196·26-s + 0.711·27-s + 0.639·28-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  =  0
Selberg data  =  $(2,\ 4030,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.252654532$
$L(\frac12)$  $\approx$  $2.252654532$
$L(\frac{3}{2})$   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 \( 1 + 2.04T + 3T^{2} \)
7 \( 1 - 3.38T + 7T^{2} \)
11 \( 1 - 4.04T + 11T^{2} \)
17 \( 1 + 0.266T + 17T^{2} \)
19 \( 1 - 0.644T + 19T^{2} \)
23 \( 1 + 2.41T + 23T^{2} \)
29 \( 1 - 3.80T + 29T^{2} \)
37 \( 1 - 7.18T + 37T^{2} \)
41 \( 1 - 9.56T + 41T^{2} \)
43 \( 1 + 3.26T + 43T^{2} \)
47 \( 1 + 5.69T + 47T^{2} \)
53 \( 1 + 4.59T + 53T^{2} \)
59 \( 1 + 7.74T + 59T^{2} \)
61 \( 1 - 1.95T + 61T^{2} \)
67 \( 1 + 0.00670T + 67T^{2} \)
71 \( 1 - 7.58T + 71T^{2} \)
73 \( 1 + 5.98T + 73T^{2} \)
79 \( 1 + 6.01T + 79T^{2} \)
83 \( 1 + 4.01T + 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 - 1.83T + 97T^{2} \)
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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

−8.221475018276795412401597746660, −7.61128265882388214563451098406, −6.72601823729123877052561279973, −6.13904624418670932676559148342, −5.42363298047928449209451343900, −4.56483211846302605083582686370, −4.31716273595086799618041567173, −3.11829652638582775914835228166, −1.84643065643365311680470427485, −0.861313946858545281333724320516, 0.861313946858545281333724320516, 1.84643065643365311680470427485, 3.11829652638582775914835228166, 4.31716273595086799618041567173, 4.56483211846302605083582686370, 5.42363298047928449209451343900, 6.13904624418670932676559148342, 6.72601823729123877052561279973, 7.61128265882388214563451098406, 8.221475018276795412401597746660

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