Properties

Degree 4
Conductor $ 2^{4} \cdot 257 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 3·8-s + 2·9-s + 4·13-s − 16-s + 4·17-s − 2·18-s − 6·25-s − 4·26-s − 12·29-s − 5·32-s − 4·34-s − 2·36-s − 16·37-s + 12·41-s + 2·49-s + 6·50-s − 4·52-s + 8·53-s + 12·58-s + 4·61-s + 7·64-s − 4·68-s + 6·72-s − 12·73-s + 16·74-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.06·8-s + 2/3·9-s + 1.10·13-s − 1/4·16-s + 0.970·17-s − 0.471·18-s − 6/5·25-s − 0.784·26-s − 2.22·29-s − 0.883·32-s − 0.685·34-s − 1/3·36-s − 2.63·37-s + 1.87·41-s + 2/7·49-s + 0.848·50-s − 0.554·52-s + 1.09·53-s + 1.57·58-s + 0.512·61-s + 7/8·64-s − 0.485·68-s + 0.707·72-s − 1.40·73-s + 1.85·74-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4112 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4112 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4112\)    =    \(2^{4} \cdot 257\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4112} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 4112,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.5440100461$
$L(\frac12)$  $\approx$  $0.5440100461$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;257\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;257\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + T + p T^{2} \)
257$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
good3$V_4$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$V_4$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$V_4$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$V_4$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
43$V_4$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
47$V_4$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$V_4$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$V_4$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$V_4$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
83$V_4$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−12.55113911913291432504173264338, −11.85008133515623933385800153806, −11.05139471712736036943594714001, −10.67459149203692243797489835816, −9.861251387498423866382206694842, −9.610791028647234859608006801501, −8.765669568456885891616139520575, −8.371288656108001150737374927218, −7.39835362300314054472468513894, −7.24477666504451448204354780334, −5.87545876078630608591993295901, −5.41426981374372084835653925495, −4.16331826282344293868388099851, −3.63582255783349937241895391884, −1.64408879948474926792686178522, 1.64408879948474926792686178522, 3.63582255783349937241895391884, 4.16331826282344293868388099851, 5.41426981374372084835653925495, 5.87545876078630608591993295901, 7.24477666504451448204354780334, 7.39835362300314054472468513894, 8.371288656108001150737374927218, 8.765669568456885891616139520575, 9.610791028647234859608006801501, 9.861251387498423866382206694842, 10.67459149203692243797489835816, 11.05139471712736036943594714001, 11.85008133515623933385800153806, 12.55113911913291432504173264338

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