Properties

Degree 4
Conductor $ 2^{2} \cdot 13^{2} \cdot 17^{2} $
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 + 4·8-s + 3·9-s − 2·13-s + 5·16-s − 3·17-s + 6·18-s + 12·19-s − 9·25-s − 4·26-s + 6·32-s − 6·34-s + 9·36-s + 24·38-s − 10·43-s + 26·47-s − 13·49-s − 18·50-s − 6·52-s + 24·53-s − 20·59-s + 7·64-s − 4·67-s − 9·68-s + 12·72-s + 36·76-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s + 9-s − 0.554·13-s + 5/4·16-s − 0.727·17-s + 1.41·18-s + 2.75·19-s − 9/5·25-s − 0.784·26-s + 1.06·32-s − 1.02·34-s + 3/2·36-s + 3.89·38-s − 1.52·43-s + 3.79·47-s − 1.85·49-s − 2.54·50-s − 0.832·52-s + 3.29·53-s − 2.60·59-s + 7/8·64-s − 0.488·67-s − 1.09·68-s + 1.41·72-s + 4.12·76-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(195364\)    =    \(2^{2} \cdot 13^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{195364} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 195364,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $4.582540933$
$L(\frac12)$  $\approx$  $4.582540933$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−9.328397420964799854901742898187, −8.601415375369587900969461268055, −7.77469056799720070910967744275, −7.45985560667588965978165906144, −7.23321315314702664404367449530, −6.68766678039395086013377186172, −5.88908450915949561478136129218, −5.68196385649536348701619256830, −5.01781591323975843915294811629, −4.61153453491182141362175201622, −3.94315686955997078455990956255, −3.56289633286702168183431548175, −2.79948963525070016890455292702, −2.11485045552833555294339924745, −1.24541420514313337396809760448, 1.24541420514313337396809760448, 2.11485045552833555294339924745, 2.79948963525070016890455292702, 3.56289633286702168183431548175, 3.94315686955997078455990956255, 4.61153453491182141362175201622, 5.01781591323975843915294811629, 5.68196385649536348701619256830, 5.88908450915949561478136129218, 6.68766678039395086013377186172, 7.23321315314702664404367449530, 7.45985560667588965978165906144, 7.77469056799720070910967744275, 8.601415375369587900969461268055, 9.328397420964799854901742898187

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