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  − 1.39·3-s − 3.54·5-s + 3.34·7-s − 1.05·9-s + 1.30·11-s − 1.98·13-s + 4.94·15-s + 7.91·17-s − 4.46·19-s − 4.66·21-s + 1.70·23-s + 7.57·25-s + 5.65·27-s + 0.208·29-s − 9.03·31-s − 1.81·33-s − 11.8·35-s − 2.45·37-s + 2.76·39-s − 3.96·41-s − 0.799·43-s + 3.74·45-s − 10.8·47-s + 4.21·49-s − 11.0·51-s + 0.665·53-s − 4.62·55-s + ⋯
L(s)  = 1  − 0.804·3-s − 1.58·5-s + 1.26·7-s − 0.352·9-s + 0.393·11-s − 0.550·13-s + 1.27·15-s + 1.91·17-s − 1.02·19-s − 1.01·21-s + 0.355·23-s + 1.51·25-s + 1.08·27-s + 0.0386·29-s − 1.62·31-s − 0.316·33-s − 2.00·35-s − 0.404·37-s + 0.442·39-s − 0.619·41-s − 0.121·43-s + 0.558·45-s − 1.57·47-s + 0.601·49-s − 1.54·51-s + 0.0914·53-s − 0.624·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8640440915$
$L(\frac12)$  $\approx$  $0.8640440915$
$L(\frac{3}{2})$   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 \( 1 + 1.39T + 3T^{2} \)
5 \( 1 + 3.54T + 5T^{2} \)
7 \( 1 - 3.34T + 7T^{2} \)
11 \( 1 - 1.30T + 11T^{2} \)
13 \( 1 + 1.98T + 13T^{2} \)
17 \( 1 - 7.91T + 17T^{2} \)
19 \( 1 + 4.46T + 19T^{2} \)
23 \( 1 - 1.70T + 23T^{2} \)
29 \( 1 - 0.208T + 29T^{2} \)
31 \( 1 + 9.03T + 31T^{2} \)
37 \( 1 + 2.45T + 37T^{2} \)
41 \( 1 + 3.96T + 41T^{2} \)
43 \( 1 + 0.799T + 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 - 0.665T + 53T^{2} \)
59 \( 1 + 12.7T + 59T^{2} \)
61 \( 1 + 1.09T + 61T^{2} \)
67 \( 1 - 14.2T + 67T^{2} \)
71 \( 1 - 3.44T + 71T^{2} \)
73 \( 1 - 15.8T + 73T^{2} \)
79 \( 1 - 8.66T + 79T^{2} \)
83 \( 1 + 13.1T + 83T^{2} \)
89 \( 1 + 1.71T + 89T^{2} \)
97 \( 1 - 17.1T + 97T^{2} \)
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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

−8.206140283466299896433468556402, −7.81162239458660576067107512444, −7.12170623564715653187730477782, −6.23327367448239701482115646068, −5.13415929380088912743874175670, −4.94497796172816419994670728479, −3.87716980434881790677605189024, −3.22014515650641875356375389046, −1.74641457664934124217715774084, −0.56228732428409182259452808687, 0.56228732428409182259452808687, 1.74641457664934124217715774084, 3.22014515650641875356375389046, 3.87716980434881790677605189024, 4.94497796172816419994670728479, 5.13415929380088912743874175670, 6.23327367448239701482115646068, 7.12170623564715653187730477782, 7.81162239458660576067107512444, 8.206140283466299896433468556402

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