Properties

Degree 2
Conductor $ 2^{4} \cdot 251 $
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  − 2.67·3-s + 4.29·5-s + 4.01·7-s + 4.16·9-s − 4.88·11-s − 3.92·13-s − 11.5·15-s + 0.345·17-s − 3.26·19-s − 10.7·21-s − 7.48·23-s + 13.4·25-s − 3.12·27-s − 10.5·29-s + 9.70·31-s + 13.0·33-s + 17.2·35-s + 4.92·37-s + 10.5·39-s + 7.39·41-s + 4.90·43-s + 17.9·45-s − 1.34·47-s + 9.14·49-s − 0.925·51-s + 1.36·53-s − 20.9·55-s + ⋯
L(s)  = 1  − 1.54·3-s + 1.92·5-s + 1.51·7-s + 1.38·9-s − 1.47·11-s − 1.08·13-s − 2.97·15-s + 0.0838·17-s − 0.749·19-s − 2.34·21-s − 1.56·23-s + 2.69·25-s − 0.601·27-s − 1.96·29-s + 1.74·31-s + 2.27·33-s + 2.91·35-s + 0.810·37-s + 1.68·39-s + 1.15·41-s + 0.748·43-s + 2.66·45-s − 0.195·47-s + 1.30·49-s − 0.129·51-s + 0.187·53-s − 2.82·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4016\)    =    \(2^{4} \cdot 251\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4016} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.596875713$
$L(\frac12)$  $\approx$  $1.596875713$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;251\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;251\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
251 \( 1 - T \)
good3 \( 1 + 2.67T + 3T^{2} \)
5 \( 1 - 4.29T + 5T^{2} \)
7 \( 1 - 4.01T + 7T^{2} \)
11 \( 1 + 4.88T + 11T^{2} \)
13 \( 1 + 3.92T + 13T^{2} \)
17 \( 1 - 0.345T + 17T^{2} \)
19 \( 1 + 3.26T + 19T^{2} \)
23 \( 1 + 7.48T + 23T^{2} \)
29 \( 1 + 10.5T + 29T^{2} \)
31 \( 1 - 9.70T + 31T^{2} \)
37 \( 1 - 4.92T + 37T^{2} \)
41 \( 1 - 7.39T + 41T^{2} \)
43 \( 1 - 4.90T + 43T^{2} \)
47 \( 1 + 1.34T + 47T^{2} \)
53 \( 1 - 1.36T + 53T^{2} \)
59 \( 1 - 0.330T + 59T^{2} \)
61 \( 1 - 9.87T + 61T^{2} \)
67 \( 1 - 11.0T + 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 - 13.6T + 73T^{2} \)
79 \( 1 - 8.91T + 79T^{2} \)
83 \( 1 + 3.12T + 83T^{2} \)
89 \( 1 + 7.28T + 89T^{2} \)
97 \( 1 - 2.84T + 97T^{2} \)
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

−8.273597926890963694141414728522, −7.70167110803242659237628922596, −6.73578662782232785380826155733, −5.92039213272661452375355266436, −5.51197544145192583821509948730, −5.00947729175569802154811952331, −4.38872185410514122944977175679, −2.28674783772858687708017794283, −2.13000070288702574958490165010, −0.77774178253262237122664530317, 0.77774178253262237122664530317, 2.13000070288702574958490165010, 2.28674783772858687708017794283, 4.38872185410514122944977175679, 5.00947729175569802154811952331, 5.51197544145192583821509948730, 5.92039213272661452375355266436, 6.73578662782232785380826155733, 7.70167110803242659237628922596, 8.273597926890963694141414728522

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