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  + 3.19·3-s − 2.92·5-s − 4.80·7-s + 7.22·9-s − 5.38·11-s + 3.67·13-s − 9.36·15-s − 1.72·17-s + 7.21·19-s − 15.3·21-s − 4.61·23-s + 3.56·25-s + 13.5·27-s + 8.38·29-s + 2.29·31-s − 17.2·33-s + 14.0·35-s − 1.44·37-s + 11.7·39-s + 2.85·41-s + 8.84·43-s − 21.1·45-s − 5.70·47-s + 16.0·49-s − 5.51·51-s + 10.4·53-s + 15.7·55-s + ⋯
L(s)  = 1  + 1.84·3-s − 1.30·5-s − 1.81·7-s + 2.40·9-s − 1.62·11-s + 1.01·13-s − 2.41·15-s − 0.418·17-s + 1.65·19-s − 3.34·21-s − 0.961·23-s + 0.713·25-s + 2.60·27-s + 1.55·29-s + 0.412·31-s − 3.00·33-s + 2.37·35-s − 0.237·37-s + 1.88·39-s + 0.446·41-s + 1.34·43-s − 3.15·45-s − 0.831·47-s + 2.29·49-s − 0.772·51-s + 1.43·53-s + 2.12·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.186925979$
$L(\frac12)$  $\approx$  $2.186925979$
$L(\frac{3}{2})$   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 \( 1 - 3.19T + 3T^{2} \)
5 \( 1 + 2.92T + 5T^{2} \)
7 \( 1 + 4.80T + 7T^{2} \)
11 \( 1 + 5.38T + 11T^{2} \)
13 \( 1 - 3.67T + 13T^{2} \)
17 \( 1 + 1.72T + 17T^{2} \)
19 \( 1 - 7.21T + 19T^{2} \)
23 \( 1 + 4.61T + 23T^{2} \)
29 \( 1 - 8.38T + 29T^{2} \)
31 \( 1 - 2.29T + 31T^{2} \)
37 \( 1 + 1.44T + 37T^{2} \)
41 \( 1 - 2.85T + 41T^{2} \)
43 \( 1 - 8.84T + 43T^{2} \)
47 \( 1 + 5.70T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 - 8.60T + 59T^{2} \)
61 \( 1 - 4.17T + 61T^{2} \)
67 \( 1 - 13.8T + 67T^{2} \)
71 \( 1 - 2.80T + 71T^{2} \)
73 \( 1 - 5.31T + 73T^{2} \)
79 \( 1 + 2.35T + 79T^{2} \)
83 \( 1 + 9.40T + 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 - 3.06T + 97T^{2} \)
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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

−8.328546449607389703759545904595, −7.902826213912923928725770145017, −7.22779559006671951181121853007, −6.58303289147154112235822379376, −5.41244875534596488952043615410, −4.11784464602442620373552700094, −3.70798322547317338947996446459, −2.94215844516425330410610797176, −2.55395186234072014398113225241, −0.75089331920601669324232082907, 0.75089331920601669324232082907, 2.55395186234072014398113225241, 2.94215844516425330410610797176, 3.70798322547317338947996446459, 4.11784464602442620373552700094, 5.41244875534596488952043615410, 6.58303289147154112235822379376, 7.22779559006671951181121853007, 7.902826213912923928725770145017, 8.328546449607389703759545904595

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