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  + 2.74·3-s + 2.14·5-s + 3.54·7-s + 4.53·9-s − 6.54·11-s + 5.92·13-s + 5.90·15-s + 3.02·17-s + 5.44·19-s + 9.74·21-s + 2.97·23-s − 0.379·25-s + 4.21·27-s − 7.08·29-s + 2.90·31-s − 17.9·33-s + 7.63·35-s + 5.80·37-s + 16.2·39-s − 8.06·41-s − 8.02·43-s + 9.74·45-s − 12.1·47-s + 5.60·49-s + 8.31·51-s − 4.41·53-s − 14.0·55-s + ⋯
L(s)  = 1  + 1.58·3-s + 0.961·5-s + 1.34·7-s + 1.51·9-s − 1.97·11-s + 1.64·13-s + 1.52·15-s + 0.734·17-s + 1.24·19-s + 2.12·21-s + 0.620·23-s − 0.0758·25-s + 0.810·27-s − 1.31·29-s + 0.521·31-s − 3.12·33-s + 1.28·35-s + 0.954·37-s + 2.60·39-s − 1.25·41-s − 1.22·43-s + 1.45·45-s − 1.77·47-s + 0.800·49-s + 1.16·51-s − 0.605·53-s − 1.89·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.926727483$
$L(\frac12)$  $\approx$  $4.926727483$
$L(\frac{3}{2})$   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 \( 1 - 2.74T + 3T^{2} \)
5 \( 1 - 2.14T + 5T^{2} \)
7 \( 1 - 3.54T + 7T^{2} \)
11 \( 1 + 6.54T + 11T^{2} \)
13 \( 1 - 5.92T + 13T^{2} \)
17 \( 1 - 3.02T + 17T^{2} \)
19 \( 1 - 5.44T + 19T^{2} \)
23 \( 1 - 2.97T + 23T^{2} \)
29 \( 1 + 7.08T + 29T^{2} \)
31 \( 1 - 2.90T + 31T^{2} \)
37 \( 1 - 5.80T + 37T^{2} \)
41 \( 1 + 8.06T + 41T^{2} \)
43 \( 1 + 8.02T + 43T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 + 4.41T + 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 - 3.39T + 61T^{2} \)
67 \( 1 + 12.2T + 67T^{2} \)
71 \( 1 - 3.79T + 71T^{2} \)
73 \( 1 + 5.43T + 73T^{2} \)
79 \( 1 - 5.75T + 79T^{2} \)
83 \( 1 - 2.88T + 83T^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 - 10.2T + 97T^{2} \)
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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

−8.236385615169811302487041186425, −7.970180010106137862591840683354, −7.42533180075282255225784171458, −6.14112291559680109764021553929, −5.32337384074146366644095973832, −4.80333192685499250165930980454, −3.46422588183715866424215522003, −2.99345817447805593788312184280, −1.93651693170343272866462024108, −1.41730891806122888634773669812, 1.41730891806122888634773669812, 1.93651693170343272866462024108, 2.99345817447805593788312184280, 3.46422588183715866424215522003, 4.80333192685499250165930980454, 5.32337384074146366644095973832, 6.14112291559680109764021553929, 7.42533180075282255225784171458, 7.970180010106137862591840683354, 8.236385615169811302487041186425

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