Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{3} \cdot 31^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·3-s + 3·4-s + 8·6-s + 7-s − 4·8-s + 6·9-s − 12·12-s − 8·13-s − 2·14-s + 5·16-s + 12·17-s − 12·18-s − 4·21-s + 16·24-s − 10·25-s + 16·26-s + 4·27-s + 3·28-s − 4·31-s − 6·32-s − 24·34-s + 18·36-s + 32·39-s + 8·42-s − 20·48-s + 49-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 3/2·4-s + 3.26·6-s + 0.377·7-s − 1.41·8-s + 2·9-s − 3.46·12-s − 2.21·13-s − 0.534·14-s + 5/4·16-s + 2.91·17-s − 2.82·18-s − 0.872·21-s + 3.26·24-s − 2·25-s + 3.13·26-s + 0.769·27-s + 0.566·28-s − 0.718·31-s − 1.06·32-s − 4.11·34-s + 3·36-s + 5.12·39-s + 1.23·42-s − 2.88·48-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1318492 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1318492 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1318492\)    =    \(2^{2} \cdot 7^{3} \cdot 31^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1318492} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 1318492,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−7.57571100088867902110310233811, −7.53412797272985138081265407651, −6.94096468302848513947365341450, −6.50565556194215905004813007391, −5.87660226535060484767774489983, −5.57928681742950427486583645839, −5.42933804945618808612902258183, −4.97433533408618033814139302887, −4.31746354726812532740296695955, −3.51037205799184365501001984019, −2.86756904365859849040052255785, −2.18321351568779845969308070860, −1.40914032002296884552019261600, −0.67731023739110916602795338706, 0, 0.67731023739110916602795338706, 1.40914032002296884552019261600, 2.18321351568779845969308070860, 2.86756904365859849040052255785, 3.51037205799184365501001984019, 4.31746354726812532740296695955, 4.97433533408618033814139302887, 5.42933804945618808612902258183, 5.57928681742950427486583645839, 5.87660226535060484767774489983, 6.50565556194215905004813007391, 6.94096468302848513947365341450, 7.53412797272985138081265407651, 7.57571100088867902110310233811

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