↑

Properties

Label 4-1280e2-1.1-c1e2-0-14
Degree $4$
Conductor $1638400$
Sign $1$
Analytic cond. $104.465$
Root an. cond. $3.19700$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 2·5-s − 4·7-s + 8·9-s + 6·13-s + 8·15-s − 6·17-s + 16·21-s + 12·23-s − 25-s − 12·27-s + 8·35-s + 6·37-s − 24·39-s − 12·43-s − 16·45-s + 12·47-s + 8·49-s + 24·51-s + 6·53-s − 16·59-s + 12·61-s − 32·63-s − 12·65-s + 12·67-s − 48·69-s + 10·73-s + ⋯
L(s)  = 1  − 2.30·3-s − 0.894·5-s − 1.51·7-s + 8/3·9-s + 1.66·13-s + 2.06·15-s − 1.45·17-s + 3.49·21-s + 2.50·23-s − 1/5·25-s − 2.30·27-s + 1.35·35-s + 0.986·37-s − 3.84·39-s − 1.82·43-s − 2.38·45-s + 1.75·47-s + 8/7·49-s + 3.36·51-s + 0.824·53-s − 2.08·59-s + 1.53·61-s − 4.03·63-s − 1.48·65-s + 1.46·67-s − 5.77·69-s + 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1638400\)    =    \(2^{16} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(104.465\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1638400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5510843942\)
\(L(\frac12)\) \(\approx\) \(0.5510843942\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
good3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.3.e_i
7$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_i
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_s
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.g_s
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.23.am_cu
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.29.a_acc
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.31.a_abu
37$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_s
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_cu
47$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_cu
53$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_s
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.59.q_ha
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.67.am_cu
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.71.a_c
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.ak_by
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_cu
89$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.89.a_agw
97$C_2^2$ \( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.97.aw_ji
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

−10.00706143645256031712785037175, −9.444564673894213322417658371415, −9.130549191581040711300163395973, −8.805673773654643069742483556352, −8.229057023724189359862427797980, −7.75025214968564446866879469944, −7.05970493174700788012366331103, −6.83535655530390151539138584253, −6.42589865469730214612895854428, −6.33932013963520786895723479845, −5.70717108921063000820608780928, −5.34513055852677239681670882525, −4.93619940605693298256242190290, −4.28479823372344926845942297995, −3.96355661100020226358907743185, −3.39426897436283552783536539144, −2.93066390695368857366513324731, −1.93794424602392892275813515642, −0.77977750204321880579753920992, −0.56826859500919761639882250958, 0.56826859500919761639882250958, 0.77977750204321880579753920992, 1.93794424602392892275813515642, 2.93066390695368857366513324731, 3.39426897436283552783536539144, 3.96355661100020226358907743185, 4.28479823372344926845942297995, 4.93619940605693298256242190290, 5.34513055852677239681670882525, 5.70717108921063000820608780928, 6.33932013963520786895723479845, 6.42589865469730214612895854428, 6.83535655530390151539138584253, 7.05970493174700788012366331103, 7.75025214968564446866879469944, 8.229057023724189359862427797980, 8.805673773654643069742483556352, 9.130549191581040711300163395973, 9.444564673894213322417658371415, 10.00706143645256031712785037175

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