Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 167 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 4·7-s − 4·11-s + 4·17-s + 4·19-s − 4·23-s − 25-s + 2·29-s − 4·31-s − 8·35-s − 12·37-s − 12·41-s + 8·43-s + 9·49-s − 14·53-s + 8·55-s + 2·59-s − 2·61-s + 4·67-s + 8·71-s + 6·73-s − 16·77-s + 14·79-s + 6·83-s − 8·85-s + 6·89-s − 8·95-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.51·7-s − 1.20·11-s + 0.970·17-s + 0.917·19-s − 0.834·23-s − 1/5·25-s + 0.371·29-s − 0.718·31-s − 1.35·35-s − 1.97·37-s − 1.87·41-s + 1.21·43-s + 9/7·49-s − 1.92·53-s + 1.07·55-s + 0.260·59-s − 0.256·61-s + 0.488·67-s + 0.949·71-s + 0.702·73-s − 1.82·77-s + 1.57·79-s + 0.658·83-s − 0.867·85-s + 0.635·89-s − 0.820·95-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 24048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 24048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(24048\)    =    \(2^{4} \cdot 3^{2} \cdot 167\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{24048} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 24048,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.650594005$
$L(\frac12)$  $\approx$  $1.650594005$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;167\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;167\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.46230309788117, −14.91031836020298, −14.34366933707070, −13.86843503411682, −13.45864967475271, −12.40777481736532, −12.17174278881113, −11.71981727114738, −10.95332881993495, −10.74537990707519, −10.00621831562651, −9.404775046375501, −8.416421065412673, −8.194761855800519, −7.619585577583873, −7.369940228807036, −6.400403539819234, −5.375901760601492, −5.216481556285921, −4.590989355514131, −3.656975308181494, −3.285308094139035, −2.175397622012844, −1.580546324929874, −0.5153285748921119, 0.5153285748921119, 1.580546324929874, 2.175397622012844, 3.285308094139035, 3.656975308181494, 4.590989355514131, 5.216481556285921, 5.375901760601492, 6.400403539819234, 7.369940228807036, 7.619585577583873, 8.194761855800519, 8.416421065412673, 9.404775046375501, 10.00621831562651, 10.74537990707519, 10.95332881993495, 11.71981727114738, 12.17174278881113, 12.40777481736532, 13.45864967475271, 13.86843503411682, 14.34366933707070, 14.91031836020298, 15.46230309788117

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