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  − 2.51·3-s − 0.472·5-s − 4.16·7-s + 3.33·9-s + 0.304·11-s + 3.93·13-s + 1.18·15-s + 5.23·17-s + 3.05·19-s + 10.4·21-s − 7.82·23-s − 4.77·25-s − 0.847·27-s − 7.79·29-s + 9.12·31-s − 0.766·33-s + 1.96·35-s − 7.45·37-s − 9.90·39-s − 5.32·41-s − 10.2·43-s − 1.57·45-s − 8.20·47-s + 10.3·49-s − 13.1·51-s − 8.29·53-s − 0.143·55-s + ⋯
L(s)  = 1  − 1.45·3-s − 0.211·5-s − 1.57·7-s + 1.11·9-s + 0.0918·11-s + 1.09·13-s + 0.306·15-s + 1.26·17-s + 0.701·19-s + 2.28·21-s − 1.63·23-s − 0.955·25-s − 0.163·27-s − 1.44·29-s + 1.63·31-s − 0.133·33-s + 0.332·35-s − 1.22·37-s − 1.58·39-s − 0.832·41-s − 1.56·43-s − 0.234·45-s − 1.19·47-s + 1.48·49-s − 1.84·51-s − 1.13·53-s − 0.0193·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5210939888$
$L(\frac12)$  $\approx$  $0.5210939888$
$L(\frac{3}{2})$   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 \( 1 + 2.51T + 3T^{2} \)
5 \( 1 + 0.472T + 5T^{2} \)
7 \( 1 + 4.16T + 7T^{2} \)
11 \( 1 - 0.304T + 11T^{2} \)
13 \( 1 - 3.93T + 13T^{2} \)
17 \( 1 - 5.23T + 17T^{2} \)
19 \( 1 - 3.05T + 19T^{2} \)
23 \( 1 + 7.82T + 23T^{2} \)
29 \( 1 + 7.79T + 29T^{2} \)
31 \( 1 - 9.12T + 31T^{2} \)
37 \( 1 + 7.45T + 37T^{2} \)
41 \( 1 + 5.32T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + 8.20T + 47T^{2} \)
53 \( 1 + 8.29T + 53T^{2} \)
59 \( 1 + 3.03T + 59T^{2} \)
61 \( 1 + 0.845T + 61T^{2} \)
67 \( 1 - 5.94T + 67T^{2} \)
71 \( 1 + 6.20T + 71T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 - 9.74T + 79T^{2} \)
83 \( 1 - 4.98T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 - 0.396T + 97T^{2} \)
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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

−8.322995174592809097836845231488, −7.62679887765525807362543099805, −6.58002135914844531897585252729, −6.26356665687670636220234712520, −5.67336510896210008075171735313, −4.89737398604161338799445599463, −3.59889303564163328176107010930, −3.41755500332368348417355932655, −1.67390349094417220839153283864, −0.44690094835506484034641559343, 0.44690094835506484034641559343, 1.67390349094417220839153283864, 3.41755500332368348417355932655, 3.59889303564163328176107010930, 4.89737398604161338799445599463, 5.67336510896210008075171735313, 6.26356665687670636220234712520, 6.58002135914844531897585252729, 7.62679887765525807362543099805, 8.322995174592809097836845231488

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