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  − 2.53·3-s + 1.50·5-s − 0.478·7-s + 3.41·9-s − 4.20·11-s + 4.32·13-s − 3.80·15-s + 2.05·17-s − 0.535·19-s + 1.21·21-s + 6.64·23-s − 2.74·25-s − 1.05·27-s + 0.258·29-s − 4.57·31-s + 10.6·33-s − 0.719·35-s − 11.8·37-s − 10.9·39-s − 5.23·41-s + 8.69·43-s + 5.13·45-s + 11.5·47-s − 6.77·49-s − 5.20·51-s + 12.2·53-s − 6.31·55-s + ⋯
L(s)  = 1  − 1.46·3-s + 0.671·5-s − 0.180·7-s + 1.13·9-s − 1.26·11-s + 1.20·13-s − 0.982·15-s + 0.498·17-s − 0.122·19-s + 0.264·21-s + 1.38·23-s − 0.548·25-s − 0.203·27-s + 0.0480·29-s − 0.822·31-s + 1.85·33-s − 0.121·35-s − 1.95·37-s − 1.75·39-s − 0.817·41-s + 1.32·43-s + 0.765·45-s + 1.68·47-s − 0.967·49-s − 0.729·51-s + 1.68·53-s − 0.851·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.052386980$
$L(\frac12)$  $\approx$  $1.052386980$
$L(\frac{3}{2})$   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 \( 1 + 2.53T + 3T^{2} \)
5 \( 1 - 1.50T + 5T^{2} \)
7 \( 1 + 0.478T + 7T^{2} \)
11 \( 1 + 4.20T + 11T^{2} \)
13 \( 1 - 4.32T + 13T^{2} \)
17 \( 1 - 2.05T + 17T^{2} \)
19 \( 1 + 0.535T + 19T^{2} \)
23 \( 1 - 6.64T + 23T^{2} \)
29 \( 1 - 0.258T + 29T^{2} \)
31 \( 1 + 4.57T + 31T^{2} \)
37 \( 1 + 11.8T + 37T^{2} \)
41 \( 1 + 5.23T + 41T^{2} \)
43 \( 1 - 8.69T + 43T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 + 3.96T + 59T^{2} \)
61 \( 1 - 4.23T + 61T^{2} \)
67 \( 1 + 14.8T + 67T^{2} \)
71 \( 1 - 16.3T + 71T^{2} \)
73 \( 1 - 6.11T + 73T^{2} \)
79 \( 1 + 10.5T + 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 - 5.39T + 89T^{2} \)
97 \( 1 - 14.0T + 97T^{2} \)
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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

−8.532657252757429093043019903652, −7.48182873469392033238730406426, −6.84398005116360025701677807459, −6.02040487588617649484182052263, −5.48534156316420181771380973322, −5.11838089606976701284544314199, −3.96333410669963484838830360952, −2.94839062310871260709808245899, −1.74126316928630625337641668160, −0.64152781743006954211110076330, 0.64152781743006954211110076330, 1.74126316928630625337641668160, 2.94839062310871260709808245899, 3.96333410669963484838830360952, 5.11838089606976701284544314199, 5.48534156316420181771380973322, 6.02040487588617649484182052263, 6.84398005116360025701677807459, 7.48182873469392033238730406426, 8.532657252757429093043019903652

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