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  + 2.85·3-s + 0.454·5-s + 2.18·7-s + 5.14·9-s + 1.15·11-s + 6.03·13-s + 1.29·15-s + 0.0535·17-s + 2.08·19-s + 6.23·21-s − 3.22·23-s − 4.79·25-s + 6.11·27-s − 1.62·29-s + 2.01·31-s + 3.30·33-s + 0.993·35-s − 3.49·37-s + 17.2·39-s + 2.85·41-s − 2.78·43-s + 2.33·45-s + 9.03·47-s − 2.21·49-s + 0.152·51-s − 4.09·53-s + 0.526·55-s + ⋯
L(s)  = 1  + 1.64·3-s + 0.203·5-s + 0.826·7-s + 1.71·9-s + 0.349·11-s + 1.67·13-s + 0.334·15-s + 0.0129·17-s + 0.479·19-s + 1.36·21-s − 0.673·23-s − 0.958·25-s + 1.17·27-s − 0.300·29-s + 0.362·31-s + 0.575·33-s + 0.167·35-s − 0.574·37-s + 2.75·39-s + 0.446·41-s − 0.423·43-s + 0.348·45-s + 1.31·47-s − 0.317·49-s + 0.0213·51-s − 0.561·53-s + 0.0710·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.587366947$
$L(\frac12)$  $\approx$  $4.587366947$
$L(\frac{3}{2})$   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 \( 1 - 2.85T + 3T^{2} \)
5 \( 1 - 0.454T + 5T^{2} \)
7 \( 1 - 2.18T + 7T^{2} \)
11 \( 1 - 1.15T + 11T^{2} \)
13 \( 1 - 6.03T + 13T^{2} \)
17 \( 1 - 0.0535T + 17T^{2} \)
19 \( 1 - 2.08T + 19T^{2} \)
23 \( 1 + 3.22T + 23T^{2} \)
29 \( 1 + 1.62T + 29T^{2} \)
31 \( 1 - 2.01T + 31T^{2} \)
37 \( 1 + 3.49T + 37T^{2} \)
41 \( 1 - 2.85T + 41T^{2} \)
43 \( 1 + 2.78T + 43T^{2} \)
47 \( 1 - 9.03T + 47T^{2} \)
53 \( 1 + 4.09T + 53T^{2} \)
59 \( 1 - 4.76T + 59T^{2} \)
61 \( 1 + 14.6T + 61T^{2} \)
67 \( 1 - 4.98T + 67T^{2} \)
71 \( 1 - 2.04T + 71T^{2} \)
73 \( 1 + 0.0937T + 73T^{2} \)
79 \( 1 - 1.92T + 79T^{2} \)
83 \( 1 + 7.96T + 83T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 - 3.00T + 97T^{2} \)
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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

−8.353662634097369171083610478452, −8.001335582674698008802641765955, −7.25592641127562168608523444052, −6.28722075580451690693416171914, −5.49952125443078286427455873601, −4.30192370835017808798004397687, −3.79174000209206429113757111520, −2.99004362511325250376539888481, −1.94154115018158437946204349055, −1.33481479269873841022474058530, 1.33481479269873841022474058530, 1.94154115018158437946204349055, 2.99004362511325250376539888481, 3.79174000209206429113757111520, 4.30192370835017808798004397687, 5.49952125443078286427455873601, 6.28722075580451690693416171914, 7.25592641127562168608523444052, 8.001335582674698008802641765955, 8.353662634097369171083610478452

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