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  − 1.55·2-s − 3-s + 0.420·4-s − 1.90·5-s + 1.55·6-s + 0.0358·7-s + 2.45·8-s + 9-s + 2.95·10-s − 4.89·11-s − 0.420·12-s + 13-s − 0.0558·14-s + 1.90·15-s − 4.66·16-s + 0.534·17-s − 1.55·18-s + 0.251·19-s − 0.798·20-s − 0.0358·21-s + 7.61·22-s + 9.05·23-s − 2.45·24-s − 1.38·25-s − 1.55·26-s − 27-s + 0.0150·28-s + ⋯
L(s)  = 1  − 1.10·2-s − 0.577·3-s + 0.210·4-s − 0.850·5-s + 0.635·6-s + 0.0135·7-s + 0.868·8-s + 0.333·9-s + 0.935·10-s − 1.47·11-s − 0.121·12-s + 0.277·13-s − 0.0149·14-s + 0.490·15-s − 1.16·16-s + 0.129·17-s − 0.366·18-s + 0.0577·19-s − 0.178·20-s − 0.00782·21-s + 1.62·22-s + 1.88·23-s − 0.501·24-s − 0.277·25-s − 0.305·26-s − 0.192·27-s + 0.00284·28-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  =  $(2,\ 4017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.3301522555$
$L(\frac12)$  $\approx$  $0.3301522555$
$L(\frac{3}{2})$   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 \( 1 + 1.55T + 2T^{2} \)
5 \( 1 + 1.90T + 5T^{2} \)
7 \( 1 - 0.0358T + 7T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
17 \( 1 - 0.534T + 17T^{2} \)
19 \( 1 - 0.251T + 19T^{2} \)
23 \( 1 - 9.05T + 23T^{2} \)
29 \( 1 - 1.09T + 29T^{2} \)
31 \( 1 + 0.0390T + 31T^{2} \)
37 \( 1 + 7.85T + 37T^{2} \)
41 \( 1 - 4.76T + 41T^{2} \)
43 \( 1 + 12.1T + 43T^{2} \)
47 \( 1 - 0.236T + 47T^{2} \)
53 \( 1 + 4.29T + 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 + 4.97T + 61T^{2} \)
67 \( 1 + 8.32T + 67T^{2} \)
71 \( 1 + 1.08T + 71T^{2} \)
73 \( 1 - 7.48T + 73T^{2} \)
79 \( 1 - 1.11T + 79T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 - 19.5T + 97T^{2} \)
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

−8.389851645990934339568269471597, −7.81382079868185163869008330744, −7.25340264116705219617431048060, −6.52193785108051169029635839615, −5.21423925590785711751645154553, −4.94436315530040162474953356777, −3.85492886219153951926741583883, −2.86711478053858177454052986651, −1.54558681469259516979150617189, −0.41520401008259682883954334463, 0.41520401008259682883954334463, 1.54558681469259516979150617189, 2.86711478053858177454052986651, 3.85492886219153951926741583883, 4.94436315530040162474953356777, 5.21423925590785711751645154553, 6.52193785108051169029635839615, 7.25340264116705219617431048060, 7.81382079868185163869008330744, 8.389851645990934339568269471597

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