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  + 3.39·3-s − 1.84·5-s + 1.40·7-s + 8.51·9-s + 4.77·11-s − 4.38·13-s − 6.25·15-s + 3.98·17-s + 3.05·19-s + 4.75·21-s + 5.20·23-s − 1.60·25-s + 18.7·27-s − 5.58·29-s + 8.38·31-s + 16.2·33-s − 2.58·35-s − 7.29·37-s − 14.8·39-s − 5.29·41-s − 8.60·43-s − 15.7·45-s − 3.56·47-s − 5.03·49-s + 13.5·51-s − 1.07·53-s − 8.80·55-s + ⋯
L(s)  = 1  + 1.95·3-s − 0.824·5-s + 0.529·7-s + 2.83·9-s + 1.44·11-s − 1.21·13-s − 1.61·15-s + 0.967·17-s + 0.700·19-s + 1.03·21-s + 1.08·23-s − 0.320·25-s + 3.60·27-s − 1.03·29-s + 1.50·31-s + 2.82·33-s − 0.436·35-s − 1.19·37-s − 2.38·39-s − 0.826·41-s − 1.31·43-s − 2.34·45-s − 0.519·47-s − 0.719·49-s + 1.89·51-s − 0.146·53-s − 1.18·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.253623396$
$L(\frac12)$  $\approx$  $4.253623396$
$L(\frac{3}{2})$   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 \( 1 - 3.39T + 3T^{2} \)
5 \( 1 + 1.84T + 5T^{2} \)
7 \( 1 - 1.40T + 7T^{2} \)
11 \( 1 - 4.77T + 11T^{2} \)
13 \( 1 + 4.38T + 13T^{2} \)
17 \( 1 - 3.98T + 17T^{2} \)
19 \( 1 - 3.05T + 19T^{2} \)
23 \( 1 - 5.20T + 23T^{2} \)
29 \( 1 + 5.58T + 29T^{2} \)
31 \( 1 - 8.38T + 31T^{2} \)
37 \( 1 + 7.29T + 37T^{2} \)
41 \( 1 + 5.29T + 41T^{2} \)
43 \( 1 + 8.60T + 43T^{2} \)
47 \( 1 + 3.56T + 47T^{2} \)
53 \( 1 + 1.07T + 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 - 1.87T + 61T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 + 6.78T + 71T^{2} \)
73 \( 1 - 13.3T + 73T^{2} \)
79 \( 1 - 16.8T + 79T^{2} \)
83 \( 1 - 2.43T + 83T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 + 4.75T + 97T^{2} \)
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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

−8.363913748947438238739382506479, −7.86161483649653048336386849326, −7.20836110796831301618838058733, −6.73681455503231313731627662530, −5.10642356379062562195057815358, −4.45514661292052807713547114481, −3.52675377933021470249463017709, −3.23109189983757941234034096785, −2.03996671597815417592486138115, −1.20888607211190496904610847376, 1.20888607211190496904610847376, 2.03996671597815417592486138115, 3.23109189983757941234034096785, 3.52675377933021470249463017709, 4.45514661292052807713547114481, 5.10642356379062562195057815358, 6.73681455503231313731627662530, 7.20836110796831301618838058733, 7.86161483649653048336386849326, 8.363913748947438238739382506479

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