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Properties

Label 4-44320-1.1-c1e2-0-1
Degree $4$
Conductor $44320$
Sign $-1$
Analytic cond. $2.82588$
Root an. cond. $1.29654$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 4·5-s + 8-s − 9-s − 4·10-s − 6·13-s + 16-s − 3·17-s − 18-s − 4·20-s + 8·25-s − 6·26-s + 32-s − 3·34-s − 36-s − 37-s − 4·40-s − 14·41-s + 4·45-s − 11·49-s + 8·50-s − 6·52-s − 4·53-s − 3·61-s + 64-s + 24·65-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.78·5-s + 0.353·8-s − 1/3·9-s − 1.26·10-s − 1.66·13-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.894·20-s + 8/5·25-s − 1.17·26-s + 0.176·32-s − 0.514·34-s − 1/6·36-s − 0.164·37-s − 0.632·40-s − 2.18·41-s + 0.596·45-s − 1.57·49-s + 1.13·50-s − 0.832·52-s − 0.549·53-s − 0.384·61-s + 1/8·64-s + 2.97·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44320 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44320 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(44320\)    =    \(2^{5} \cdot 5 \cdot 277\)
Sign: $-1$
Analytic conductor: \(2.82588\)
Root analytic conductor: \(1.29654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 44320,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( 1 - T \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 3 T + p T^{2} ) \)
277$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 14 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.7.a_l
11$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.11.a_aq
13$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.g_ba
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.19.a_d
23$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \) 2.23.a_n
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.a_ax
31$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.31.a_q
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.37.b_bg
41$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.o_es
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.43.a_cc
47$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.47.a_ak
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.e_bu
59$C_2^2$ \( 1 + 97 T^{2} + p^{2} T^{4} \) 2.59.a_dt
61$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.d_ca
67$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \) 2.67.a_abf
71$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.71.a_ar
73$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.73.ak_go
79$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \) 2.79.a_aef
83$C_2^2$ \( 1 - 138 T^{2} + p^{2} T^{4} \) 2.83.a_afi
89$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 - 15 T + p T^{2} ) \) 2.89.abg_qr
97$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.r_gu
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.978001109985468344931045471392, −9.508994638742077000322275275092, −8.720528103790899353839679023753, −8.254227824377826787170235701949, −7.76654948566917035757894587134, −7.30489602534198156896809288752, −6.80141846546954801603887352322, −6.28331881356758074112157463337, −5.23496382409112442979456026575, −4.84336663246060735257288282519, −4.35597437353318628506145750261, −3.54904721195067800475988388385, −3.06867131677246475420079393771, −2.06996045115149143338334510501, 0, 2.06996045115149143338334510501, 3.06867131677246475420079393771, 3.54904721195067800475988388385, 4.35597437353318628506145750261, 4.84336663246060735257288282519, 5.23496382409112442979456026575, 6.28331881356758074112157463337, 6.80141846546954801603887352322, 7.30489602534198156896809288752, 7.76654948566917035757894587134, 8.254227824377826787170235701949, 8.720528103790899353839679023753, 9.508994638742077000322275275092, 9.978001109985468344931045471392

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