Properties

Degree 2
Conductor $ 2^{4} \cdot 251 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.26·3-s − 4.10·5-s + 1.61·7-s + 7.63·9-s − 5.52·11-s + 3.26·13-s + 13.3·15-s − 4.72·17-s + 3.75·19-s − 5.25·21-s − 4.65·23-s + 11.8·25-s − 15.1·27-s − 3.54·29-s − 5.76·31-s + 18.0·33-s − 6.61·35-s − 0.102·37-s − 10.6·39-s + 10.2·41-s − 2.60·43-s − 31.3·45-s − 6.85·47-s − 4.40·49-s + 15.4·51-s + 3.77·53-s + 22.6·55-s + ⋯
L(s)  = 1  − 1.88·3-s − 1.83·5-s + 0.609·7-s + 2.54·9-s − 1.66·11-s + 0.904·13-s + 3.45·15-s − 1.14·17-s + 0.861·19-s − 1.14·21-s − 0.971·23-s + 2.36·25-s − 2.91·27-s − 0.658·29-s − 1.03·31-s + 3.13·33-s − 1.11·35-s − 0.0168·37-s − 1.70·39-s + 1.59·41-s − 0.396·43-s − 4.67·45-s − 1.00·47-s − 0.628·49-s + 2.15·51-s + 0.518·53-s + 3.05·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.1599508187$
$L(\frac12)$  $\approx$  $0.1599508187$
$L(\frac{3}{2})$   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 \( 1 + 3.26T + 3T^{2} \)
5 \( 1 + 4.10T + 5T^{2} \)
7 \( 1 - 1.61T + 7T^{2} \)
11 \( 1 + 5.52T + 11T^{2} \)
13 \( 1 - 3.26T + 13T^{2} \)
17 \( 1 + 4.72T + 17T^{2} \)
19 \( 1 - 3.75T + 19T^{2} \)
23 \( 1 + 4.65T + 23T^{2} \)
29 \( 1 + 3.54T + 29T^{2} \)
31 \( 1 + 5.76T + 31T^{2} \)
37 \( 1 + 0.102T + 37T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 + 2.60T + 43T^{2} \)
47 \( 1 + 6.85T + 47T^{2} \)
53 \( 1 - 3.77T + 53T^{2} \)
59 \( 1 + 5.75T + 59T^{2} \)
61 \( 1 + 9.91T + 61T^{2} \)
67 \( 1 + 13.8T + 67T^{2} \)
71 \( 1 + 15.0T + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 - 6.36T + 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 - 1.56T + 89T^{2} \)
97 \( 1 + 6.16T + 97T^{2} \)
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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

−8.114353582245910799410700786988, −7.57164860533146066027806174630, −7.14651297425982487863335563043, −6.06191933378700610959762023530, −5.48139447274228151421822538722, −4.56459904924428354139473798669, −4.31659522155290157631956338049, −3.19337412415024042613079694065, −1.56618666361384379902507205548, −0.25761067189979796360831373952, 0.25761067189979796360831373952, 1.56618666361384379902507205548, 3.19337412415024042613079694065, 4.31659522155290157631956338049, 4.56459904924428354139473798669, 5.48139447274228151421822538722, 6.06191933378700610959762023530, 7.14651297425982487863335563043, 7.57164860533146066027806174630, 8.114353582245910799410700786988

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