Properties

Degree 4
Conductor $ 2^{7} \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-s + 4-s + 8-s − 6·9-s + 16-s − 12·17-s − 6·18-s + 16·23-s − 6·25-s − 2·31-s + 32-s − 12·34-s − 6·36-s − 12·41-s + 16·46-s − 16·47-s − 14·49-s − 6·50-s − 2·62-s + 64-s − 12·68-s + 16·71-s − 6·72-s + 20·73-s − 16·79-s + 27·81-s − 12·82-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 2·9-s + 1/4·16-s − 2.91·17-s − 1.41·18-s + 3.33·23-s − 6/5·25-s − 0.359·31-s + 0.176·32-s − 2.05·34-s − 36-s − 1.87·41-s + 2.35·46-s − 2.33·47-s − 2·49-s − 0.848·50-s − 0.254·62-s + 1/8·64-s − 1.45·68-s + 1.89·71-s − 0.707·72-s + 2.34·73-s − 1.80·79-s + 3·81-s − 1.32·82-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 123008 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 123008 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(123008\)    =    \(2^{7} \cdot 31^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{123008} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 123008,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

Imaginary part of the first few zeros on the critical line

−9.056498267458590117851162621111, −8.714468530691354459730603161713, −8.307477897920151884753308617841, −7.76857292337562945056886541715, −6.81859577538004307044633877685, −6.64449933506524691025726128618, −6.32858579420662158347624019611, −5.39201619892659079650657036147, −5.00958996845543469784857559007, −4.74203912193695697715499891246, −3.69780406318668437933515679302, −3.16188554299090604324575557253, −2.60552830936859898841087279326, −1.84479362740755814713596587903, 0, 1.84479362740755814713596587903, 2.60552830936859898841087279326, 3.16188554299090604324575557253, 3.69780406318668437933515679302, 4.74203912193695697715499891246, 5.00958996845543469784857559007, 5.39201619892659079650657036147, 6.32858579420662158347624019611, 6.64449933506524691025726128618, 6.81859577538004307044633877685, 7.76857292337562945056886541715, 8.307477897920151884753308617841, 8.714468530691354459730603161713, 9.056498267458590117851162621111

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