↑

Properties

Label 4-8379e2-1.1-c1e2-0-0
Degree $4$
Conductor $70207641$
Sign $1$
Analytic cond. $4476.50$
Root an. cond. $8.17964$
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-s + 2·5-s + 4·10-s + 2·11-s + 16-s − 4·17-s − 2·19-s + 2·20-s + 4·22-s − 2·23-s + 25-s + 4·29-s − 12·31-s − 2·32-s − 8·34-s + 8·37-s − 4·38-s − 4·41-s − 14·43-s + 2·44-s − 4·46-s − 18·47-s + 2·50-s + 4·53-s + 4·55-s + 8·58-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s + 0.894·5-s + 1.26·10-s + 0.603·11-s + 1/4·16-s − 0.970·17-s − 0.458·19-s + 0.447·20-s + 0.852·22-s − 0.417·23-s + 1/5·25-s + 0.742·29-s − 2.15·31-s − 0.353·32-s − 1.37·34-s + 1.31·37-s − 0.648·38-s − 0.624·41-s − 2.13·43-s + 0.301·44-s − 0.589·46-s − 2.62·47-s + 0.282·50-s + 0.549·53-s + 0.539·55-s + 1.05·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70207641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70207641 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(70207641\)    =    \(3^{4} \cdot 7^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(4476.50\)
Root analytic conductor: \(8.17964\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 70207641,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4045170777\)
\(L(\frac12)\) \(\approx\) \(0.4045170777\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
7 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.2.ac_d
5$D_{4}$ \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_d
11$D_{4}$ \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_p
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$C_4$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_g
23$D_{4}$ \( 1 + 2 T + 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_bn
29$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_cc
31$D_{4}$ \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.31.m_dm
37$D_{4}$ \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.37.ai_de
41$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_cc
43$D_{4}$ \( 1 + 14 T + 103 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.43.o_dz
47$D_{4}$ \( 1 + 18 T + 167 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.47.s_gl
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.53.ae_eg
59$D_{4}$ \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.59.q_gs
61$D_{4}$ \( 1 + 18 T + 171 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.61.s_gp
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$D_{4}$ \( 1 + 20 T + 210 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.71.u_ic
73$D_{4}$ \( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.73.ac_t
79$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.79.aq_hi
83$D_{4}$ \( 1 + 14 T + 207 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.83.o_hz
89$D_{4}$ \( 1 + 16 T + 234 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.89.q_ja
97$D_{4}$ \( 1 - 28 T + 382 T^{2} - 28 p T^{3} + p^{2} T^{4} \) 2.97.abc_os
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

−7.83263779532065215188139260766, −7.39186616028546697067545586928, −7.38339003687213147439767350464, −6.61456317085116609378301022167, −6.49441932491779124870811787795, −6.18534755255517844545005576251, −5.80122191701016133853148485879, −5.63826504828734904478674258120, −4.97447331974842749874321414018, −4.79828078333765794003161177689, −4.47582617202134937864740921630, −4.32325371252366613096075432847, −3.70221403746767035556992547579, −3.33567118835517075670017424799, −3.06424817070466360044516943899, −2.63082680657280336023198856740, −1.78494248570683923105349552154, −1.70140781380202816137512596075, −1.46228871865551032818498935455, −0.10175573901711212033374417786, 0.10175573901711212033374417786, 1.46228871865551032818498935455, 1.70140781380202816137512596075, 1.78494248570683923105349552154, 2.63082680657280336023198856740, 3.06424817070466360044516943899, 3.33567118835517075670017424799, 3.70221403746767035556992547579, 4.32325371252366613096075432847, 4.47582617202134937864740921630, 4.79828078333765794003161177689, 4.97447331974842749874321414018, 5.63826504828734904478674258120, 5.80122191701016133853148485879, 6.18534755255517844545005576251, 6.49441932491779124870811787795, 6.61456317085116609378301022167, 7.38339003687213147439767350464, 7.39186616028546697067545586928, 7.83263779532065215188139260766

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