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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 1.60·5-s − 6-s + 2.84·7-s − 8-s + 9-s + 1.60·10-s − 4.57·11-s + 12-s + 13-s − 2.84·14-s − 1.60·15-s + 16-s + 6.96·17-s − 18-s + 5.10·19-s − 1.60·20-s + 2.84·21-s + 4.57·22-s − 7.39·23-s − 24-s − 2.41·25-s − 26-s + 27-s + 2.84·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.719·5-s − 0.408·6-s + 1.07·7-s − 0.353·8-s + 0.333·9-s + 0.508·10-s − 1.37·11-s + 0.288·12-s + 0.277·13-s − 0.759·14-s − 0.415·15-s + 0.250·16-s + 1.68·17-s − 0.235·18-s + 1.17·19-s − 0.359·20-s + 0.620·21-s + 0.974·22-s − 1.54·23-s − 0.204·24-s − 0.482·25-s − 0.196·26-s + 0.192·27-s + 0.537·28-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
13 \( 1 - T \)
103 \( 1 + T \)
good5 \( 1 + 1.60T + 5T^{2} \)
7 \( 1 - 2.84T + 7T^{2} \)
11 \( 1 + 4.57T + 11T^{2} \)
17 \( 1 - 6.96T + 17T^{2} \)
19 \( 1 - 5.10T + 19T^{2} \)
23 \( 1 + 7.39T + 23T^{2} \)
29 \( 1 + 7.17T + 29T^{2} \)
31 \( 1 - 4.44T + 31T^{2} \)
37 \( 1 + 0.817T + 37T^{2} \)
41 \( 1 + 11.4T + 41T^{2} \)
43 \( 1 - 4.69T + 43T^{2} \)
47 \( 1 + 6.40T + 47T^{2} \)
53 \( 1 + 8.74T + 53T^{2} \)
59 \( 1 - 0.862T + 59T^{2} \)
61 \( 1 + 5.94T + 61T^{2} \)
67 \( 1 + 5.12T + 67T^{2} \)
71 \( 1 - 5.91T + 71T^{2} \)
73 \( 1 + 1.48T + 73T^{2} \)
79 \( 1 + 16.1T + 79T^{2} \)
83 \( 1 - 7.18T + 83T^{2} \)
89 \( 1 - 13.9T + 89T^{2} \)
97 \( 1 + 2.58T + 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.71070864717467808484079229076, −7.37534090627859664465157274645, −6.08223772544658348639859820663, −5.38989757459047350134606971398, −4.71532280745393403547082138235, −3.62960973354934711364524432731, −3.14074611147801204189167812982, −2.05185962892504425154692456490, −1.32221705223917583617736199467, 0, 1.32221705223917583617736199467, 2.05185962892504425154692456490, 3.14074611147801204189167812982, 3.62960973354934711364524432731, 4.71532280745393403547082138235, 5.38989757459047350134606971398, 6.08223772544658348639859820663, 7.37534090627859664465157274645, 7.71070864717467808484079229076

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