Properties

Degree 2
Conductor $ 2^{4} \cdot 5077 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 4·5-s + 4·7-s + 6·9-s + 6·11-s − 4·13-s − 12·15-s − 4·17-s + 7·19-s + 12·21-s + 6·23-s + 11·25-s + 9·27-s − 6·29-s + 2·31-s + 18·33-s − 16·35-s − 12·39-s + 8·43-s − 24·45-s + 9·47-s + 9·49-s − 12·51-s − 9·53-s − 24·55-s + 21·57-s + 11·59-s + ⋯
L(s)  = 1  + 1.73·3-s − 1.78·5-s + 1.51·7-s + 2·9-s + 1.80·11-s − 1.10·13-s − 3.09·15-s − 0.970·17-s + 1.60·19-s + 2.61·21-s + 1.25·23-s + 11/5·25-s + 1.73·27-s − 1.11·29-s + 0.359·31-s + 3.13·33-s − 2.70·35-s − 1.92·39-s + 1.21·43-s − 3.57·45-s + 1.31·47-s + 9/7·49-s − 1.68·51-s − 1.23·53-s − 3.23·55-s + 2.78·57-s + 1.43·59-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 81232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 81232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(81232\)    =    \(2^{4} \cdot 5077\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{81232} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 81232,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.922193072$
$L(\frac12)$  $\approx$  $5.922193072$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5077\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5077\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5077 \( 1 + T \)
good3 \( 1 - p T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.24210642202560, −13.74043991742789, −12.98224781638525, −12.47075890717160, −11.90566042292306, −11.47592504697466, −11.25997877488675, −10.58115220179354, −9.642730044374383, −9.245886687225486, −8.840792293204737, −8.473565046413400, −7.767351125915900, −7.535895889623422, −7.158049812917592, −6.696866960871701, −5.435419181975811, −4.735985009314551, −4.381863722451139, −3.887864718899586, −3.440739333779044, −2.760332665264791, −2.111595985041348, −1.321534985496399, −0.7844861948204286, 0.7844861948204286, 1.321534985496399, 2.111595985041348, 2.760332665264791, 3.440739333779044, 3.887864718899586, 4.381863722451139, 4.735985009314551, 5.435419181975811, 6.696866960871701, 7.158049812917592, 7.535895889623422, 7.767351125915900, 8.473565046413400, 8.840792293204737, 9.245886687225486, 9.642730044374383, 10.58115220179354, 11.25997877488675, 11.47592504697466, 11.90566042292306, 12.47075890717160, 12.98224781638525, 13.74043991742789, 14.24210642202560

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