↑

Properties

Label 4-440e2-1.1-c1e2-0-23
Degree $4$
Conductor $193600$
Sign $1$
Analytic cond. $12.3441$
Root an. cond. $1.87441$
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  + 2·2-s + 4·3-s + 2·4-s + 2·5-s + 8·6-s + 6·9-s + 4·10-s + 8·12-s + 4·13-s + 8·15-s − 4·16-s + 12·18-s + 4·20-s − 25-s + 8·26-s − 4·27-s + 16·30-s − 20·31-s − 8·32-s + 12·36-s − 12·37-s + 16·39-s + 12·41-s + 16·43-s + 12·45-s − 16·48-s − 2·49-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.30·3-s + 4-s + 0.894·5-s + 3.26·6-s + 2·9-s + 1.26·10-s + 2.30·12-s + 1.10·13-s + 2.06·15-s − 16-s + 2.82·18-s + 0.894·20-s − 1/5·25-s + 1.56·26-s − 0.769·27-s + 2.92·30-s − 3.59·31-s − 1.41·32-s + 2·36-s − 1.97·37-s + 2.56·39-s + 1.87·41-s + 2.43·43-s + 1.78·45-s − 2.30·48-s − 2/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(193600\)    =    \(2^{6} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(12.3441\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 193600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.206010790\)
\(L(\frac12)\) \(\approx\) \(8.206010790\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
11$C_2$ \( 1 + T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.3.ae_k
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.19.a_abm
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.23.a_s
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.29.a_aw
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.31.u_gg
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.53.au_hy
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.59.a_ady
61$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.61.a_adi
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.67.ae_fi
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.73.a_afm
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.79.ai_gs
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.83.ai_ha
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2^2$ \( 1 - 178 T^{2} + p^{2} T^{4} \) 2.97.a_agw
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

−11.19744098434246178364628988664, −11.04750566946265956898419747228, −10.56901315246732434203167428060, −9.748165685748620782993350597035, −9.289420086052832864180153774762, −9.046476258338687210453402771405, −8.826940832710163719419807406936, −8.277885475323207834490316020153, −7.60003854061187031265143154649, −7.30510608396538496138823056769, −6.69041192135671957809394819018, −5.85041025025590157942578484142, −5.62748779000104612935375022788, −5.29745790883100398287185830005, −4.07036747342335262142454222992, −3.76199013689032357209954790650, −3.59364184994705351780019173249, −2.66901441165047728458130933206, −2.32353444322024915853090546013, −1.75629605492916321007445059957, 1.75629605492916321007445059957, 2.32353444322024915853090546013, 2.66901441165047728458130933206, 3.59364184994705351780019173249, 3.76199013689032357209954790650, 4.07036747342335262142454222992, 5.29745790883100398287185830005, 5.62748779000104612935375022788, 5.85041025025590157942578484142, 6.69041192135671957809394819018, 7.30510608396538496138823056769, 7.60003854061187031265143154649, 8.277885475323207834490316020153, 8.826940832710163719419807406936, 9.046476258338687210453402771405, 9.289420086052832864180153774762, 9.748165685748620782993350597035, 10.56901315246732434203167428060, 11.04750566946265956898419747228, 11.19744098434246178364628988664

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