Properties

Degree 4
Conductor $ 13^{2} \cdot 37^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4·4-s − 3·9-s − 8·12-s − 4·13-s + 12·16-s + 12·17-s + 12·23-s − 10·25-s − 14·27-s − 12·29-s + 12·36-s − 8·39-s + 16·43-s + 24·48-s − 13·49-s + 24·51-s + 16·52-s − 6·53-s + 16·61-s − 32·64-s − 48·68-s + 24·69-s − 20·75-s − 20·79-s − 4·81-s − 24·87-s + ⋯
L(s)  = 1  + 1.15·3-s − 2·4-s − 9-s − 2.30·12-s − 1.10·13-s + 3·16-s + 2.91·17-s + 2.50·23-s − 2·25-s − 2.69·27-s − 2.22·29-s + 2·36-s − 1.28·39-s + 2.43·43-s + 3.46·48-s − 1.85·49-s + 3.36·51-s + 2.21·52-s − 0.824·53-s + 2.04·61-s − 4·64-s − 5.82·68-s + 2.88·69-s − 2.30·75-s − 2.25·79-s − 4/9·81-s − 2.57·87-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 231361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 231361 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(231361\)    =    \(13^{2} \cdot 37^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{231361} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 231361,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{13,\;37\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{13,\;37\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad13$C_2$ \( 1 + 4 T + p T^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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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

−9.032345851694468357832029856913, −8.241330915774091826817607809326, −7.932252712474312694875118416259, −7.59911177067371416247669368567, −7.33591996649962500060513363229, −5.97965532485801959040073464192, −5.52163728124810212778032448117, −5.44973416215471530225317007593, −4.84323437999328576885334632783, −3.96753607282613353734743159883, −3.50910294340479626549928321873, −3.22077100740487422350904766970, −2.49170215186160867754656573885, −1.24474461137231690537182545671, 0, 1.24474461137231690537182545671, 2.49170215186160867754656573885, 3.22077100740487422350904766970, 3.50910294340479626549928321873, 3.96753607282613353734743159883, 4.84323437999328576885334632783, 5.44973416215471530225317007593, 5.52163728124810212778032448117, 5.97965532485801959040073464192, 7.33591996649962500060513363229, 7.59911177067371416247669368567, 7.932252712474312694875118416259, 8.241330915774091826817607809326, 9.032345851694468357832029856913

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