Properties

Degree 4
Conductor $ 2^{2} \cdot 29^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 2·5-s − 4·7-s + 5·9-s − 2·13-s + 16-s − 2·20-s − 12·23-s − 7·25-s + 4·28-s + 10·29-s − 8·35-s − 5·36-s + 10·45-s − 2·49-s + 2·52-s − 2·53-s + 20·59-s − 20·63-s − 64-s − 4·65-s + 16·67-s − 16·71-s + 2·80-s + 16·81-s + 28·83-s + 8·91-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.894·5-s − 1.51·7-s + 5/3·9-s − 0.554·13-s + 1/4·16-s − 0.447·20-s − 2.50·23-s − 7/5·25-s + 0.755·28-s + 1.85·29-s − 1.35·35-s − 5/6·36-s + 1.49·45-s − 2/7·49-s + 0.277·52-s − 0.274·53-s + 2.60·59-s − 2.51·63-s − 1/8·64-s − 0.496·65-s + 1.95·67-s − 1.89·71-s + 0.223·80-s + 16/9·81-s + 3.07·83-s + 0.838·91-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3364\)    =    \(2^{2} \cdot 29^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3364} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 3364,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.7251012065$
$L(\frac12)$  $\approx$  $0.7251012065$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;29\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;29\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + T^{2} \)
29$C_2$ \( 1 - 10 T + p T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 157 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.79223661837676310690463501944, −12.19159594137950504415190357871, −11.84645776206177216181121347730, −10.42310480291414364991838362568, −10.11355450344949753919883085106, −9.606145269789884566896974019170, −9.574900076543553179030084851064, −8.304437480803866732936226297063, −7.69276887016559049573974177703, −6.66752135726560757641321157411, −6.35874181585401900380951870048, −5.46741702281727274032318267978, −4.37322746476470901250864601121, −3.67787809784757201819998183332, −2.15684754175300937699526679967, 2.15684754175300937699526679967, 3.67787809784757201819998183332, 4.37322746476470901250864601121, 5.46741702281727274032318267978, 6.35874181585401900380951870048, 6.66752135726560757641321157411, 7.69276887016559049573974177703, 8.304437480803866732936226297063, 9.574900076543553179030084851064, 9.606145269789884566896974019170, 10.11355450344949753919883085106, 10.42310480291414364991838362568, 11.84645776206177216181121347730, 12.19159594137950504415190357871, 12.79223661837676310690463501944

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