↑

Properties

Label 4-2442e2-1.1-c1e2-0-0
Degree $4$
Conductor $5963364$
Sign $1$
Analytic cond. $380.229$
Root an. cond. $4.41582$
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  − 2·3-s + 4-s + 3·9-s + 11-s − 2·12-s + 16-s − 2·23-s − 10·25-s − 4·27-s − 12·31-s − 2·33-s + 3·36-s − 2·37-s + 44-s − 12·47-s − 2·48-s − 5·49-s − 2·53-s + 64-s + 4·67-s + 4·69-s + 20·75-s + 5·81-s − 6·89-s − 2·92-s + 24·93-s − 20·97-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s + 9-s + 0.301·11-s − 0.577·12-s + 1/4·16-s − 0.417·23-s − 2·25-s − 0.769·27-s − 2.15·31-s − 0.348·33-s + 1/2·36-s − 0.328·37-s + 0.150·44-s − 1.75·47-s − 0.288·48-s − 5/7·49-s − 0.274·53-s + 1/8·64-s + 0.488·67-s + 0.481·69-s + 2.30·75-s + 5/9·81-s − 0.635·89-s − 0.208·92-s + 2.48·93-s − 2.03·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5963364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5963364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5963364\)    =    \(2^{2} \cdot 3^{2} \cdot 11^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(380.229\)
Root analytic conductor: \(4.41582\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5963364,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5500936517\)
\(L(\frac12)\) \(\approx\) \(0.5500936517\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( ( 1 + T )^{2} \)
11$C_2$ \( 1 - T + p T^{2} \)
37$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.13.a_z
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.a_z
19$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.19.a_bd
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.23.c_bv
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.a_bq
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.31.m_du
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.a_acg
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.47.m_fa
53$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.53.c_ed
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.a_eo
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.67.ae_fi
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.73.a_fh
79$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.79.a_abm
83$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.83.a_dh
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.89.g_hf
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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

−7.07433016066728121125499972487, −6.99093330071255894275494820065, −6.26030991441211289089148573592, −5.95009651878852125475078071763, −5.85671815365648243578194470848, −5.21904617077696602305759220700, −4.93990077281798148757107408401, −4.38634303924562361149721823813, −3.81933209527513554138512512650, −3.61646225738397093676153018687, −3.02120025281117424297881430460, −2.14827435614165614269081834365, −1.81998707337995438672490638604, −1.34568190088063630717234001359, −0.26605500028990923666648464644, 0.26605500028990923666648464644, 1.34568190088063630717234001359, 1.81998707337995438672490638604, 2.14827435614165614269081834365, 3.02120025281117424297881430460, 3.61646225738397093676153018687, 3.81933209527513554138512512650, 4.38634303924562361149721823813, 4.93990077281798148757107408401, 5.21904617077696602305759220700, 5.85671815365648243578194470848, 5.95009651878852125475078071763, 6.26030991441211289089148573592, 6.99093330071255894275494820065, 7.07433016066728121125499972487

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