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  + 3.22·3-s + 3.75·5-s − 2.77·7-s + 7.38·9-s + 3.10·11-s + 4.47·13-s + 12.1·15-s + 4.42·17-s − 7.35·19-s − 8.93·21-s − 6.95·23-s + 9.13·25-s + 14.1·27-s − 3.27·29-s − 1.68·31-s + 10.0·33-s − 10.4·35-s − 9.33·37-s + 14.4·39-s + 1.88·41-s + 12.6·43-s + 27.7·45-s + 6.26·47-s + 0.686·49-s + 14.2·51-s − 3.80·53-s + 11.6·55-s + ⋯
L(s)  = 1  + 1.86·3-s + 1.68·5-s − 1.04·7-s + 2.46·9-s + 0.936·11-s + 1.24·13-s + 3.12·15-s + 1.07·17-s − 1.68·19-s − 1.95·21-s − 1.44·23-s + 1.82·25-s + 2.72·27-s − 0.607·29-s − 0.303·31-s + 1.74·33-s − 1.76·35-s − 1.53·37-s + 2.31·39-s + 0.294·41-s + 1.93·43-s + 4.14·45-s + 0.913·47-s + 0.0981·49-s + 1.99·51-s − 0.522·53-s + 1.57·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.304785878$
$L(\frac12)$  $\approx$  $5.304785878$
$L(\frac{3}{2})$   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 \( 1 - 3.22T + 3T^{2} \)
5 \( 1 - 3.75T + 5T^{2} \)
7 \( 1 + 2.77T + 7T^{2} \)
11 \( 1 - 3.10T + 11T^{2} \)
13 \( 1 - 4.47T + 13T^{2} \)
17 \( 1 - 4.42T + 17T^{2} \)
19 \( 1 + 7.35T + 19T^{2} \)
23 \( 1 + 6.95T + 23T^{2} \)
29 \( 1 + 3.27T + 29T^{2} \)
31 \( 1 + 1.68T + 31T^{2} \)
37 \( 1 + 9.33T + 37T^{2} \)
41 \( 1 - 1.88T + 41T^{2} \)
43 \( 1 - 12.6T + 43T^{2} \)
47 \( 1 - 6.26T + 47T^{2} \)
53 \( 1 + 3.80T + 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 + 2.70T + 61T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 - 1.87T + 71T^{2} \)
73 \( 1 + 9.50T + 73T^{2} \)
79 \( 1 - 10.6T + 79T^{2} \)
83 \( 1 + 6.37T + 83T^{2} \)
89 \( 1 - 0.850T + 89T^{2} \)
97 \( 1 + 5.86T + 97T^{2} \)
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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

−8.729695883620009979685848491829, −7.888786855817550112843371612024, −6.96215931719911209466522420134, −6.18162305054219203047290791524, −5.86232284540377738598222651040, −4.26693922541225498539266024242, −3.66866480683862287644506219091, −2.91432326464540546807334430018, −1.99735166062483697203395058041, −1.45459059593748622659370423384, 1.45459059593748622659370423384, 1.99735166062483697203395058041, 2.91432326464540546807334430018, 3.66866480683862287644506219091, 4.26693922541225498539266024242, 5.86232284540377738598222651040, 6.18162305054219203047290791524, 6.96215931719911209466522420134, 7.888786855817550112843371612024, 8.729695883620009979685848491829

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