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Properties

Label 4-825e2-1.1-c1e2-0-6
Degree $4$
Conductor $680625$
Sign $1$
Analytic cond. $43.3972$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 4-s + 4·6-s + 2·7-s + 3·9-s − 2·11-s − 2·12-s − 4·14-s + 16-s − 2·17-s − 6·18-s + 10·19-s − 4·21-s + 4·22-s − 2·23-s − 4·27-s + 2·28-s + 8·29-s + 2·32-s + 4·33-s + 4·34-s + 3·36-s + 6·37-s − 20·38-s − 2·41-s + 8·42-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 1/2·4-s + 1.63·6-s + 0.755·7-s + 9-s − 0.603·11-s − 0.577·12-s − 1.06·14-s + 1/4·16-s − 0.485·17-s − 1.41·18-s + 2.29·19-s − 0.872·21-s + 0.852·22-s − 0.417·23-s − 0.769·27-s + 0.377·28-s + 1.48·29-s + 0.353·32-s + 0.696·33-s + 0.685·34-s + 1/2·36-s + 0.986·37-s − 3.24·38-s − 0.312·41-s + 1.23·42-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(680625\)    =    \(3^{2} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(43.3972\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 680625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6264090067\)
\(L(\frac12)\) \(\approx\) \(0.6264090067\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.2.c_d
7$D_{4}$ \( 1 - 2 T + 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_n
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$D_{4}$ \( 1 + 2 T + 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_bh
19$D_{4}$ \( 1 - 10 T + 61 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.19.ak_cj
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.23.c_bv
29$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.29.ai_co
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.31.a_ak
37$D_{4}$ \( 1 - 6 T + 75 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_cx
41$D_{4}$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.41.c_ap
43$D_{4}$ \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.43.am_dm
47$D_{4}$ \( 1 + 2 T + 23 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.47.c_x
53$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.53.e_da
59$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.59.aw_jf
61$D_{4}$ \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.61.am_fu
67$D_{4}$ \( 1 - 12 T + 138 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.67.am_fi
71$D_{4}$ \( 1 - 10 T + 159 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.71.ak_gd
73$D_{4}$ \( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.73.am_gs
79$D_{4}$ \( 1 - 18 T + 221 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.79.as_in
83$D_{4}$ \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.83.i_eg
89$D_{4}$ \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.89.e_fu
97$D_{4}$ \( 1 + 6 T + 195 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.97.g_hn
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.09766746946000514003095168972, −10.06303731155086646313729724025, −9.619820724121131911116471516685, −9.411609650044072762521810834062, −8.663637355351682074465559656083, −8.362312826759208787159508141675, −7.84774838436040119835415118124, −7.75036923751843003731406402542, −7.04181707120955560905574405964, −6.64330462173694054216256407012, −6.17566666897077187845548492209, −5.52179103088108848114589450846, −5.14076392050016555645541912914, −4.89606359484530109473100527393, −4.17695700964742228333548377812, −3.55880570075872032781970268036, −2.71587874174246573609519224571, −2.06395509747984882662455571235, −0.925745063711959340691621415458, −0.77242910565716435963065767883, 0.77242910565716435963065767883, 0.925745063711959340691621415458, 2.06395509747984882662455571235, 2.71587874174246573609519224571, 3.55880570075872032781970268036, 4.17695700964742228333548377812, 4.89606359484530109473100527393, 5.14076392050016555645541912914, 5.52179103088108848114589450846, 6.17566666897077187845548492209, 6.64330462173694054216256407012, 7.04181707120955560905574405964, 7.75036923751843003731406402542, 7.84774838436040119835415118124, 8.362312826759208787159508141675, 8.663637355351682074465559656083, 9.411609650044072762521810834062, 9.619820724121131911116471516685, 10.06303731155086646313729724025, 10.09766746946000514003095168972

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