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  + 1.40·3-s + 2.59·5-s − 1.10·7-s − 1.02·9-s + 5.45·11-s − 2.10·13-s + 3.64·15-s − 1.93·17-s + 4.08·19-s − 1.55·21-s + 2.90·23-s + 1.73·25-s − 5.65·27-s + 5.78·29-s + 1.15·31-s + 7.66·33-s − 2.86·35-s − 1.95·37-s − 2.96·39-s + 2.94·41-s + 6.50·43-s − 2.66·45-s + 11.0·47-s − 5.78·49-s − 2.71·51-s + 0.0920·53-s + 14.1·55-s + ⋯
L(s)  = 1  + 0.811·3-s + 1.16·5-s − 0.417·7-s − 0.342·9-s + 1.64·11-s − 0.584·13-s + 0.941·15-s − 0.469·17-s + 0.938·19-s − 0.338·21-s + 0.605·23-s + 0.347·25-s − 1.08·27-s + 1.07·29-s + 0.206·31-s + 1.33·33-s − 0.484·35-s − 0.321·37-s − 0.474·39-s + 0.459·41-s + 0.991·43-s − 0.397·45-s + 1.61·47-s − 0.825·49-s − 0.380·51-s + 0.0126·53-s + 1.91·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.298416987$
$L(\frac12)$  $\approx$  $3.298416987$
$L(\frac{3}{2})$   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 \( 1 - 1.40T + 3T^{2} \)
5 \( 1 - 2.59T + 5T^{2} \)
7 \( 1 + 1.10T + 7T^{2} \)
11 \( 1 - 5.45T + 11T^{2} \)
13 \( 1 + 2.10T + 13T^{2} \)
17 \( 1 + 1.93T + 17T^{2} \)
19 \( 1 - 4.08T + 19T^{2} \)
23 \( 1 - 2.90T + 23T^{2} \)
29 \( 1 - 5.78T + 29T^{2} \)
31 \( 1 - 1.15T + 31T^{2} \)
37 \( 1 + 1.95T + 37T^{2} \)
41 \( 1 - 2.94T + 41T^{2} \)
43 \( 1 - 6.50T + 43T^{2} \)
47 \( 1 - 11.0T + 47T^{2} \)
53 \( 1 - 0.0920T + 53T^{2} \)
59 \( 1 + 1.65T + 59T^{2} \)
61 \( 1 - 5.04T + 61T^{2} \)
67 \( 1 + 8.55T + 67T^{2} \)
71 \( 1 - 7.42T + 71T^{2} \)
73 \( 1 + 10.9T + 73T^{2} \)
79 \( 1 - 7.18T + 79T^{2} \)
83 \( 1 - 6.37T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 + 2.75T + 97T^{2} \)
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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

−8.755949089856895415918353090086, −7.72007279202862775208707664458, −6.92100259463334186879943416679, −6.25895149147912833211937254152, −5.62094431105460623505668908086, −4.63150511029859664386149035044, −3.67509115375160407341723550049, −2.84942987297462058278158839652, −2.12902293862787225918694722189, −1.05409168097639532888011641531, 1.05409168097639532888011641531, 2.12902293862787225918694722189, 2.84942987297462058278158839652, 3.67509115375160407341723550049, 4.63150511029859664386149035044, 5.62094431105460623505668908086, 6.25895149147912833211937254152, 6.92100259463334186879943416679, 7.72007279202862775208707664458, 8.755949089856895415918353090086

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