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Properties

Label 4-50240-1.1-c1e2-0-2
Degree $4$
Conductor $50240$
Sign $-1$
Analytic cond. $3.20334$
Root an. cond. $1.33783$
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  − 4·5-s + 4·9-s − 5·13-s − 3·17-s + 8·25-s − 3·29-s − 11·37-s − 3·41-s − 16·45-s + 11·49-s − 18·53-s + 61-s + 20·65-s − 14·73-s + 7·81-s + 12·85-s − 6·89-s + 4·97-s + 4·109-s − 27·113-s − 20·117-s + 5·121-s − 5·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1.78·5-s + 4/3·9-s − 1.38·13-s − 0.727·17-s + 8/5·25-s − 0.557·29-s − 1.80·37-s − 0.468·41-s − 2.38·45-s + 11/7·49-s − 2.47·53-s + 0.128·61-s + 2.48·65-s − 1.63·73-s + 7/9·81-s + 1.30·85-s − 0.635·89-s + 0.406·97-s + 0.383·109-s − 2.53·113-s − 1.84·117-s + 5/11·121-s − 0.447·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50240 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50240 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50240\)    =    \(2^{6} \cdot 5 \cdot 157\)
Sign: $-1$
Analytic conductor: \(3.20334\)
Root analytic conductor: \(1.33783\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 50240,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 3 T + p T^{2} ) \)
157$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 13 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.a_al
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.11.a_af
13$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.f_be
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.19.a_ax
23$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.23.a_t
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.29.d_cg
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.37.l_dy
41$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.d_cm
43$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \) 2.43.a_abp
47$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \) 2.47.a_cy
53$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.s_gw
59$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.59.a_ack
61$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.ab_m
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.71.a_adf
73$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.73.o_gd
79$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.79.a_bu
83$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.83.a_ach
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.g_gw
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.ae_cc
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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

−9.858045743819311509868228647444, −9.363322067316703649862164052961, −8.722831102242672382131073441412, −8.254588582692908432219818281974, −7.54604886543374070697521879810, −7.32320405952787478354876052341, −6.98023618165182953508518599255, −6.26974785868887323548907532839, −5.20201013528890253294389484113, −4.75138125005888345353256840231, −4.17022076032998284080462283973, −3.71855858734373324052450879085, −2.84795404790166873418769614350, −1.71857017341068937488066981904, 0, 1.71857017341068937488066981904, 2.84795404790166873418769614350, 3.71855858734373324052450879085, 4.17022076032998284080462283973, 4.75138125005888345353256840231, 5.20201013528890253294389484113, 6.26974785868887323548907532839, 6.98023618165182953508518599255, 7.32320405952787478354876052341, 7.54604886543374070697521879810, 8.254588582692908432219818281974, 8.722831102242672382131073441412, 9.363322067316703649862164052961, 9.858045743819311509868228647444

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