Properties

Degree 4
Conductor $ 2^{2} \cdot 17^{3} $
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 − 2·9-s + 4·13-s + 5·16-s − 17-s − 4·18-s − 8·19-s − 10·25-s + 8·26-s + 6·32-s − 2·34-s − 6·36-s − 16·38-s + 16·43-s + 2·49-s − 20·50-s + 12·52-s − 12·53-s + 7·64-s + 16·67-s − 3·68-s − 8·72-s − 24·76-s − 5·81-s + 32·86-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s − 2/3·9-s + 1.10·13-s + 5/4·16-s − 0.242·17-s − 0.942·18-s − 1.83·19-s − 2·25-s + 1.56·26-s + 1.06·32-s − 0.342·34-s − 36-s − 2.59·38-s + 2.43·43-s + 2/7·49-s − 2.82·50-s + 1.66·52-s − 1.64·53-s + 7/8·64-s + 1.95·67-s − 0.363·68-s − 0.942·72-s − 2.75·76-s − 5/9·81-s + 3.45·86-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 19652 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 19652 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(19652\)    =    \(2^{2} \cdot 17^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{19652} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 19652,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.450942393$
$L(\frac12)$  $\approx$  $2.450942393$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

Imaginary part of the first few zeros on the critical line

−10.88013907095950671991603611029, −10.84894668716717132123408330664, −9.942505353869395091050894698746, −9.286071988264369821451329617999, −8.551880873534616105011482646500, −8.107939322961323795970351474484, −7.46682025356819819578371193029, −6.65330731261113455462227374582, −5.99911855171913780844071238816, −5.93029628134094737560230707913, −4.99927116218852259459902565637, −4.05951678114838271873159817135, −3.90229547122613769308947697962, −2.77156487822388402999564657844, −1.95202314698375464079211503776, 1.95202314698375464079211503776, 2.77156487822388402999564657844, 3.90229547122613769308947697962, 4.05951678114838271873159817135, 4.99927116218852259459902565637, 5.93029628134094737560230707913, 5.99911855171913780844071238816, 6.65330731261113455462227374582, 7.46682025356819819578371193029, 8.107939322961323795970351474484, 8.551880873534616105011482646500, 9.286071988264369821451329617999, 9.942505353869395091050894698746, 10.84894668716717132123408330664, 10.88013907095950671991603611029

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