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  + 0.347·3-s − 1.66·5-s − 3.92·7-s − 2.87·9-s − 6.50·11-s − 3.17·13-s − 0.579·15-s + 3.56·17-s − 8.45·19-s − 1.36·21-s + 4.47·23-s − 2.22·25-s − 2.04·27-s − 9.13·29-s + 4.27·31-s − 2.26·33-s + 6.53·35-s + 0.328·37-s − 1.10·39-s + 8.75·41-s + 4.27·43-s + 4.79·45-s − 5.78·47-s + 8.37·49-s + 1.24·51-s + 13.1·53-s + 10.8·55-s + ⋯
L(s)  = 1  + 0.200·3-s − 0.745·5-s − 1.48·7-s − 0.959·9-s − 1.96·11-s − 0.879·13-s − 0.149·15-s + 0.864·17-s − 1.93·19-s − 0.297·21-s + 0.932·23-s − 0.444·25-s − 0.393·27-s − 1.69·29-s + 0.766·31-s − 0.393·33-s + 1.10·35-s + 0.0540·37-s − 0.176·39-s + 1.36·41-s + 0.651·43-s + 0.715·45-s − 0.843·47-s + 1.19·49-s + 0.173·51-s + 1.81·53-s + 1.46·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.1466462537$
$L(\frac12)$  $\approx$  $0.1466462537$
$L(\frac{3}{2})$   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 \( 1 - 0.347T + 3T^{2} \)
5 \( 1 + 1.66T + 5T^{2} \)
7 \( 1 + 3.92T + 7T^{2} \)
11 \( 1 + 6.50T + 11T^{2} \)
13 \( 1 + 3.17T + 13T^{2} \)
17 \( 1 - 3.56T + 17T^{2} \)
19 \( 1 + 8.45T + 19T^{2} \)
23 \( 1 - 4.47T + 23T^{2} \)
29 \( 1 + 9.13T + 29T^{2} \)
31 \( 1 - 4.27T + 31T^{2} \)
37 \( 1 - 0.328T + 37T^{2} \)
41 \( 1 - 8.75T + 41T^{2} \)
43 \( 1 - 4.27T + 43T^{2} \)
47 \( 1 + 5.78T + 47T^{2} \)
53 \( 1 - 13.1T + 53T^{2} \)
59 \( 1 + 6.76T + 59T^{2} \)
61 \( 1 + 9.71T + 61T^{2} \)
67 \( 1 + 11.0T + 67T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 - 13.0T + 73T^{2} \)
79 \( 1 + 4.45T + 79T^{2} \)
83 \( 1 - 7.31T + 83T^{2} \)
89 \( 1 - 0.686T + 89T^{2} \)
97 \( 1 + 7.21T + 97T^{2} \)
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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

−8.334033005036708094340261176260, −7.70373989450473641191475705459, −7.20272292778802961096018234048, −6.09201381758113390308992061300, −5.62149161638668019186262669776, −4.64449109196106056289500744547, −3.68623625141692407420905609902, −2.85437036747831927620198962712, −2.42471066536631314436574735566, −0.19543783440016964273481579782, 0.19543783440016964273481579782, 2.42471066536631314436574735566, 2.85437036747831927620198962712, 3.68623625141692407420905609902, 4.64449109196106056289500744547, 5.62149161638668019186262669776, 6.09201381758113390308992061300, 7.20272292778802961096018234048, 7.70373989450473641191475705459, 8.334033005036708094340261176260

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