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  − 3.15·3-s + 4.04·5-s − 3.52·7-s + 6.93·9-s + 5.53·11-s + 0.479·13-s − 12.7·15-s − 7.91·17-s + 5.32·19-s + 11.0·21-s + 3.98·23-s + 11.3·25-s − 12.3·27-s + 4.58·29-s + 0.0567·31-s − 17.4·33-s − 14.2·35-s − 1.74·37-s − 1.51·39-s + 4.51·41-s + 11.5·43-s + 28.0·45-s − 11.2·47-s + 5.39·49-s + 24.9·51-s + 4.14·53-s + 22.3·55-s + ⋯
L(s)  = 1  − 1.81·3-s + 1.80·5-s − 1.33·7-s + 2.31·9-s + 1.66·11-s + 0.132·13-s − 3.29·15-s − 1.92·17-s + 1.22·19-s + 2.42·21-s + 0.830·23-s + 2.27·25-s − 2.38·27-s + 0.851·29-s + 0.0101·31-s − 3.03·33-s − 2.40·35-s − 0.286·37-s − 0.241·39-s + 0.705·41-s + 1.76·43-s + 4.17·45-s − 1.64·47-s + 0.770·49-s + 3.49·51-s + 0.570·53-s + 3.01·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.408978265$
$L(\frac12)$  $\approx$  $1.408978265$
$L(\frac{3}{2})$   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 \( 1 + 3.15T + 3T^{2} \)
5 \( 1 - 4.04T + 5T^{2} \)
7 \( 1 + 3.52T + 7T^{2} \)
11 \( 1 - 5.53T + 11T^{2} \)
13 \( 1 - 0.479T + 13T^{2} \)
17 \( 1 + 7.91T + 17T^{2} \)
19 \( 1 - 5.32T + 19T^{2} \)
23 \( 1 - 3.98T + 23T^{2} \)
29 \( 1 - 4.58T + 29T^{2} \)
31 \( 1 - 0.0567T + 31T^{2} \)
37 \( 1 + 1.74T + 37T^{2} \)
41 \( 1 - 4.51T + 41T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 - 4.14T + 53T^{2} \)
59 \( 1 - 4.02T + 59T^{2} \)
61 \( 1 + 12.3T + 61T^{2} \)
67 \( 1 - 5.97T + 67T^{2} \)
71 \( 1 + 5.10T + 71T^{2} \)
73 \( 1 - 2.78T + 73T^{2} \)
79 \( 1 - 2.61T + 79T^{2} \)
83 \( 1 + 1.48T + 83T^{2} \)
89 \( 1 + 6.81T + 89T^{2} \)
97 \( 1 + 5.45T + 97T^{2} \)
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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

−8.908247681354894868436166879628, −7.00685394624339033814310432317, −6.70820312270005225393273848395, −6.23816684361790781970458824363, −5.74139086368586418384594032626, −4.91108185707604607719087097081, −4.12407279676052863200581934749, −2.85486081613543697878092015388, −1.62944086604856653989600856132, −0.795605398493410347806565803314, 0.795605398493410347806565803314, 1.62944086604856653989600856132, 2.85486081613543697878092015388, 4.12407279676052863200581934749, 4.91108185707604607719087097081, 5.74139086368586418384594032626, 6.23816684361790781970458824363, 6.70820312270005225393273848395, 7.00685394624339033814310432317, 8.908247681354894868436166879628

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