Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{3} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s + 7-s − 4·8-s − 2·9-s − 2·14-s + 5·16-s + 4·18-s − 10·25-s + 3·28-s − 12·29-s − 6·32-s − 6·36-s + 4·37-s + 16·43-s + 49-s + 20·50-s + 12·53-s − 4·56-s + 24·58-s − 2·63-s + 7·64-s − 8·67-s + 8·72-s − 8·74-s + 16·79-s − 5·81-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.377·7-s − 1.41·8-s − 2/3·9-s − 0.534·14-s + 5/4·16-s + 0.942·18-s − 2·25-s + 0.566·28-s − 2.22·29-s − 1.06·32-s − 36-s + 0.657·37-s + 2.43·43-s + 1/7·49-s + 2.82·50-s + 1.64·53-s − 0.534·56-s + 3.15·58-s − 0.251·63-s + 7/8·64-s − 0.977·67-s + 0.942·72-s − 0.929·74-s + 1.80·79-s − 5/9·81-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1372 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1372 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1372\)    =    \(2^{2} \cdot 7^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1372} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1372,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3308765761$
$L(\frac12)$  $\approx$  $0.3308765761$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( 1 - T \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−13.87946529878520861319815819980, −13.15728760628296679290378410395, −12.30526099005271094386573698884, −11.59222775413431567626281153330, −11.23136141438460806904855801814, −10.57696390398788315518055643332, −9.765547119459919407856461234632, −9.205467104844532229657218112988, −8.624667079434487291694904422005, −7.57571100088867902110310233811, −7.55563465371540871894331307120, −6.12986038892916091786195943044, −5.57928681742950427486583645839, −3.87985229263682105648190826671, −2.25140513775369021989127014661, 2.25140513775369021989127014661, 3.87985229263682105648190826671, 5.57928681742950427486583645839, 6.12986038892916091786195943044, 7.55563465371540871894331307120, 7.57571100088867902110310233811, 8.624667079434487291694904422005, 9.205467104844532229657218112988, 9.765547119459919407856461234632, 10.57696390398788315518055643332, 11.23136141438460806904855801814, 11.59222775413431567626281153330, 12.30526099005271094386573698884, 13.15728760628296679290378410395, 13.87946529878520861319815819980

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