Properties

Degree 4
Conductor $ 2^{2} \cdot 17^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

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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 − 6·17-s + 4·18-s − 8·19-s + 2·25-s − 8·26-s − 6·32-s + 12·34-s − 6·36-s + 16·38-s − 8·43-s + 14·49-s − 4·50-s + 12·52-s + 12·53-s + 24·59-s + 7·64-s − 8·67-s − 18·68-s + 8·72-s − 24·76-s − 5·81-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 − 1.45·17-s + 0.942·18-s − 1.83·19-s + 2/5·25-s − 1.56·26-s − 1.06·32-s + 2.05·34-s − 36-s + 2.59·38-s − 1.21·43-s + 2·49-s − 0.565·50-s + 1.66·52-s + 1.64·53-s + 3.12·59-s + 7/8·64-s − 0.977·67-s − 2.18·68-s + 0.942·72-s − 2.75·76-s − 5/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1156\)    =    \(2^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1156} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 1156,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.3093859603\)
\(L(\frac12)\)  \(\approx\)  \(0.3093859603\)
\(L(\frac{3}{2})\)   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 $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + T )^{2} \)
17$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
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$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−14.12719507516391158951572154901, −13.23246082696141745797793564410, −12.84960809136784896974761597774, −11.62104328433788920008879166606, −11.48138420354830170197930020217, −10.44767684414425161642458569644, −10.42251693501845675043344465453, −9.112705956177541863647570903426, −8.517394393692955508869058389691, −8.484102564279125782220274863224, −7.10604028804496855298145497397, −6.53747231897222211021137211021, −5.68070638546849464095536524868, −4.05533330791451947168279880511, −2.37714604467739077916333919542, 2.37714604467739077916333919542, 4.05533330791451947168279880511, 5.68070638546849464095536524868, 6.53747231897222211021137211021, 7.10604028804496855298145497397, 8.484102564279125782220274863224, 8.517394393692955508869058389691, 9.112705956177541863647570903426, 10.42251693501845675043344465453, 10.44767684414425161642458569644, 11.48138420354830170197930020217, 11.62104328433788920008879166606, 12.84960809136784896974761597774, 13.23246082696141745797793564410, 14.12719507516391158951572154901

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