Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{4} $
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-s + 2·3-s + 2·6-s − 8-s + 3·9-s − 8·13-s − 16-s − 6·17-s + 3·18-s − 2·19-s − 2·24-s + 5·25-s − 8·26-s + 10·27-s − 12·29-s + 4·31-s − 6·34-s − 2·37-s − 2·38-s − 16·39-s + 12·41-s + 16·43-s + 12·47-s − 2·48-s + 5·50-s − 12·51-s − 6·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 0.816·6-s − 0.353·8-s + 9-s − 2.21·13-s − 1/4·16-s − 1.45·17-s + 0.707·18-s − 0.458·19-s − 0.408·24-s + 25-s − 1.56·26-s + 1.92·27-s − 2.22·29-s + 0.718·31-s − 1.02·34-s − 0.328·37-s − 0.324·38-s − 2.56·39-s + 1.87·41-s + 2.43·43-s + 1.75·47-s − 0.288·48-s + 0.707·50-s − 1.68·51-s − 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(9604\)    =    \(2^{2} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{98} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 9604,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.68244$
$L(\frac12)$  $\approx$  $1.68244$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−14.04145501722198037479383964324, −14.00599460194856240485032040018, −12.89897144252259646211093501170, −12.85516300122271479135410455197, −12.44533047287876342504227065817, −11.74368042716781372037397857686, −10.84462218398902176922880365263, −10.60537085923855477135106016776, −9.630199992252261854634945255997, −9.241789904092050155683227766971, −8.904163825273414140536527508881, −8.083333619699282901638240448876, −7.25391899305612172511439282965, −7.16393847545088947450776580615, −6.12510237641322476348159870666, −5.23488858771214743865463879013, −4.42448063842485448054280162476, −4.12098676425324769033179756617, −2.71904573650174489385286764559, −2.41749920092248519897743465838, 2.41749920092248519897743465838, 2.71904573650174489385286764559, 4.12098676425324769033179756617, 4.42448063842485448054280162476, 5.23488858771214743865463879013, 6.12510237641322476348159870666, 7.16393847545088947450776580615, 7.25391899305612172511439282965, 8.083333619699282901638240448876, 8.904163825273414140536527508881, 9.241789904092050155683227766971, 9.630199992252261854634945255997, 10.60537085923855477135106016776, 10.84462218398902176922880365263, 11.74368042716781372037397857686, 12.44533047287876342504227065817, 12.85516300122271479135410455197, 12.89897144252259646211093501170, 14.00599460194856240485032040018, 14.04145501722198037479383964324

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