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  + 1.15·3-s − 4.09·5-s + 0.621·7-s − 1.66·9-s − 0.176·11-s − 6.62·13-s − 4.73·15-s − 4.87·17-s + 6.67·19-s + 0.718·21-s − 1.95·23-s + 11.7·25-s − 5.39·27-s − 4.79·29-s + 3.71·31-s − 0.204·33-s − 2.54·35-s + 6.22·37-s − 7.65·39-s + 1.84·41-s + 6.89·43-s + 6.81·45-s + 0.545·47-s − 6.61·49-s − 5.63·51-s − 4.30·53-s + 0.723·55-s + ⋯
L(s)  = 1  + 0.667·3-s − 1.83·5-s + 0.234·7-s − 0.554·9-s − 0.0532·11-s − 1.83·13-s − 1.22·15-s − 1.18·17-s + 1.53·19-s + 0.156·21-s − 0.406·23-s + 2.35·25-s − 1.03·27-s − 0.891·29-s + 0.666·31-s − 0.0355·33-s − 0.430·35-s + 1.02·37-s − 1.22·39-s + 0.288·41-s + 1.05·43-s + 1.01·45-s + 0.0795·47-s − 0.944·49-s − 0.789·51-s − 0.591·53-s + 0.0975·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9269796099$
$L(\frac12)$  $\approx$  $0.9269796099$
$L(\frac{3}{2})$   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 \( 1 - 1.15T + 3T^{2} \)
5 \( 1 + 4.09T + 5T^{2} \)
7 \( 1 - 0.621T + 7T^{2} \)
11 \( 1 + 0.176T + 11T^{2} \)
13 \( 1 + 6.62T + 13T^{2} \)
17 \( 1 + 4.87T + 17T^{2} \)
19 \( 1 - 6.67T + 19T^{2} \)
23 \( 1 + 1.95T + 23T^{2} \)
29 \( 1 + 4.79T + 29T^{2} \)
31 \( 1 - 3.71T + 31T^{2} \)
37 \( 1 - 6.22T + 37T^{2} \)
41 \( 1 - 1.84T + 41T^{2} \)
43 \( 1 - 6.89T + 43T^{2} \)
47 \( 1 - 0.545T + 47T^{2} \)
53 \( 1 + 4.30T + 53T^{2} \)
59 \( 1 + 11.0T + 59T^{2} \)
61 \( 1 + 3.06T + 61T^{2} \)
67 \( 1 - 9.29T + 67T^{2} \)
71 \( 1 - 6.86T + 71T^{2} \)
73 \( 1 - 5.09T + 73T^{2} \)
79 \( 1 - 7.28T + 79T^{2} \)
83 \( 1 - 18.1T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 - 8.30T + 97T^{2} \)
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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

−8.124617864639902889890675194746, −7.79101168139561346344531421637, −7.38728080890983022523867494313, −6.43513198237761692628426893130, −5.15981891989092126431418175173, −4.61148561737165257785538511987, −3.77621272111550057015793839375, −3.00404374352961898501404347763, −2.26966536817348726753482445218, −0.50312947777833023468855912453, 0.50312947777833023468855912453, 2.26966536817348726753482445218, 3.00404374352961898501404347763, 3.77621272111550057015793839375, 4.61148561737165257785538511987, 5.15981891989092126431418175173, 6.43513198237761692628426893130, 7.38728080890983022523867494313, 7.79101168139561346344531421637, 8.124617864639902889890675194746

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