↑

Properties

Label 4-23552-1.1-c1e2-0-7
Degree $4$
Conductor $23552$
Sign $-1$
Analytic cond. $1.50169$
Root an. cond. $1.10699$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 5·9-s − 3·17-s − 5·23-s − 25-s + 5·31-s − 15·41-s + 3·47-s + 7·49-s + 20·63-s − 6·71-s + 10·73-s − 7·79-s + 16·81-s − 3·89-s + 10·97-s − 10·103-s + 3·113-s + 12·119-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 15·153-s + ⋯
L(s)  = 1  − 1.51·7-s − 5/3·9-s − 0.727·17-s − 1.04·23-s − 1/5·25-s + 0.898·31-s − 2.34·41-s + 0.437·47-s + 49-s + 2.51·63-s − 0.712·71-s + 1.17·73-s − 0.787·79-s + 16/9·81-s − 0.317·89-s + 1.01·97-s − 0.985·103-s + 0.282·113-s + 1.10·119-s + 8/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.21·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23552 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23552 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(23552\)    =    \(2^{10} \cdot 23\)
Sign: $-1$
Analytic conductor: \(1.50169\)
Root analytic conductor: \(1.10699\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 23552,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
23$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 6 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.e_j
11$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.11.a_ai
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.19.a_au
29$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.29.a_abg
31$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.af_bw
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.37.a_abm
41$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.41.p_fg
43$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \) 2.43.a_acq
47$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.47.ad_dq
53$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.53.a_ax
59$C_2^2$ \( 1 + 91 T^{2} + p^{2} T^{4} \) 2.59.a_dn
61$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.61.a_af
67$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.67.a_bu
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.g_fm
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.73.ak_dm
79$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.h_fu
83$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \) 2.83.a_ca
89$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.d_ge
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.97.ak_fi
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.35023934045051728196092745967, −9.935030293868057985956598022891, −9.444916473342122923961944903425, −8.750394435265591935757505078883, −8.480211289056841010861757031595, −7.84285348791469575452137265703, −6.94948043135094331643814920372, −6.48381527190025923699869948748, −5.99555553969489693972072563490, −5.45657064867732047332952533028, −4.57584877100759231197948710668, −3.61311432495519144124358950117, −3.08008002117098331205491472104, −2.26827636064597550610394227730, 0, 2.26827636064597550610394227730, 3.08008002117098331205491472104, 3.61311432495519144124358950117, 4.57584877100759231197948710668, 5.45657064867732047332952533028, 5.99555553969489693972072563490, 6.48381527190025923699869948748, 6.94948043135094331643814920372, 7.84285348791469575452137265703, 8.480211289056841010861757031595, 8.750394435265591935757505078883, 9.444916473342122923961944903425, 9.935030293868057985956598022891, 10.35023934045051728196092745967

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