Properties

Degree 2
Conductor $ 3 \cdot 13 \cdot 103 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 0.858·2-s − 3-s − 1.26·4-s − 4.02·5-s + 0.858·6-s + 1.69·7-s + 2.80·8-s + 9-s + 3.45·10-s + 4.28·11-s + 1.26·12-s + 13-s − 1.45·14-s + 4.02·15-s + 0.120·16-s + 7.37·17-s − 0.858·18-s + 0.247·19-s + 5.08·20-s − 1.69·21-s − 3.68·22-s + 7.02·23-s − 2.80·24-s + 11.2·25-s − 0.858·26-s − 27-s − 2.13·28-s + ⋯
L(s)  = 1  − 0.607·2-s − 0.577·3-s − 0.631·4-s − 1.80·5-s + 0.350·6-s + 0.639·7-s + 0.990·8-s + 0.333·9-s + 1.09·10-s + 1.29·11-s + 0.364·12-s + 0.277·13-s − 0.387·14-s + 1.03·15-s + 0.0301·16-s + 1.78·17-s − 0.202·18-s + 0.0568·19-s + 1.13·20-s − 0.368·21-s − 0.785·22-s + 1.46·23-s − 0.571·24-s + 2.24·25-s − 0.168·26-s − 0.192·27-s − 0.403·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4017\)    =    \(3 \cdot 13 \cdot 103\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4017} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8887908627$
$L(\frac12)$  $\approx$  $0.8887908627$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;13,\;103\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;13,\;103\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
13 \( 1 - T \)
103 \( 1 + T \)
good2 \( 1 + 0.858T + 2T^{2} \)
5 \( 1 + 4.02T + 5T^{2} \)
7 \( 1 - 1.69T + 7T^{2} \)
11 \( 1 - 4.28T + 11T^{2} \)
17 \( 1 - 7.37T + 17T^{2} \)
19 \( 1 - 0.247T + 19T^{2} \)
23 \( 1 - 7.02T + 23T^{2} \)
29 \( 1 + 2.37T + 29T^{2} \)
31 \( 1 - 7.06T + 31T^{2} \)
37 \( 1 + 0.651T + 37T^{2} \)
41 \( 1 - 4.11T + 41T^{2} \)
43 \( 1 + 4.16T + 43T^{2} \)
47 \( 1 - 2.62T + 47T^{2} \)
53 \( 1 + 7.05T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 - 9.22T + 61T^{2} \)
67 \( 1 - 0.887T + 67T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 - 8.42T + 73T^{2} \)
79 \( 1 - 5.27T + 79T^{2} \)
83 \( 1 - 11.2T + 83T^{2} \)
89 \( 1 - 0.789T + 89T^{2} \)
97 \( 1 + 14.1T + 97T^{2} \)
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

−8.248455225199817323973293848312, −7.903409464791945592577068117568, −7.23349175646669621160856500485, −6.43982565552016961398640764502, −5.18282410656816844010444562347, −4.69466898526352522961025726595, −3.86926309163997259327109081590, −3.33427687063631898103581171065, −1.27724393392296465274660646952, −0.75428285198665737373365684782, 0.75428285198665737373365684782, 1.27724393392296465274660646952, 3.33427687063631898103581171065, 3.86926309163997259327109081590, 4.69466898526352522961025726595, 5.18282410656816844010444562347, 6.43982565552016961398640764502, 7.23349175646669621160856500485, 7.903409464791945592577068117568, 8.248455225199817323973293848312

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