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Properties

Label 4-4598e2-1.1-c1e2-0-15
Degree $4$
Conductor $21141604$
Sign $1$
Analytic cond. $1348.00$
Root an. cond. $6.05930$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 3·4-s − 5-s − 6·6-s + 7-s + 4·8-s + 4·9-s − 2·10-s − 9·12-s + 3·13-s + 2·14-s + 3·15-s + 5·16-s + 2·17-s + 8·18-s − 2·19-s − 3·20-s − 3·21-s − 10·23-s − 12·24-s − 6·25-s + 6·26-s − 6·27-s + 3·28-s + 13·29-s + 6·30-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.73·3-s + 3/2·4-s − 0.447·5-s − 2.44·6-s + 0.377·7-s + 1.41·8-s + 4/3·9-s − 0.632·10-s − 2.59·12-s + 0.832·13-s + 0.534·14-s + 0.774·15-s + 5/4·16-s + 0.485·17-s + 1.88·18-s − 0.458·19-s − 0.670·20-s − 0.654·21-s − 2.08·23-s − 2.44·24-s − 6/5·25-s + 1.17·26-s − 1.15·27-s + 0.566·28-s + 2.41·29-s + 1.09·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21141604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21141604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21141604\)    =    \(2^{2} \cdot 11^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1348.00\)
Root analytic conductor: \(6.05930\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 21141604,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good3$C_4$ \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.3.d_f
5$D_{4}$ \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_h
7$D_{4}$ \( 1 - T + 11 T^{2} - p T^{3} + p^{2} T^{4} \) 2.7.ab_l
13$D_{4}$ \( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.13.ad_z
17$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_w
23$D_{4}$ \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.23.k_cg
29$D_{4}$ \( 1 - 13 T + 97 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.29.an_dt
31$D_{4}$ \( 1 + 3 T + 61 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.31.d_cj
37$D_{4}$ \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_ba
41$D_{4}$ \( 1 - 3 T + 55 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_cd
43$D_{4}$ \( 1 + 13 T + 99 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.43.n_dv
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.47.m_fa
53$D_{4}$ \( 1 + 14 T + 142 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.53.o_fm
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$D_{4}$ \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_di
67$D_{4}$ \( 1 - 5 T + 59 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.67.af_ch
71$D_{4}$ \( 1 + 23 T + 271 T^{2} + 23 p T^{3} + p^{2} T^{4} \) 2.71.x_kl
73$D_{4}$ \( 1 + 2 T + 134 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_fe
79$D_{4}$ \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.79.k_co
83$D_{4}$ \( 1 + 25 T + 319 T^{2} + 25 p T^{3} + p^{2} T^{4} \) 2.83.z_mh
89$D_{4}$ \( 1 - 2 T + 166 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.89.ac_gk
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.97.aq_jy
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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

−8.056280428413773612815340095489, −7.69851203475497045286173525712, −7.24077775462987778537210071402, −6.85098900121693001421745282720, −6.27618429580222546328848748775, −6.25515810304723313731381613557, −5.89686889359369810282120351182, −5.77147233262827214693079606726, −4.98774941147231156275822313531, −4.88095705277131245675559300256, −4.58830060096391877825894587037, −4.09656461779126087443721896265, −3.73736176759141151028476772946, −3.26205160426145124720208221811, −2.94909210675309568214675799174, −2.09476324187039181284348221417, −1.48989443334990104529489706532, −1.45591921190020855854057917926, 0, 0, 1.45591921190020855854057917926, 1.48989443334990104529489706532, 2.09476324187039181284348221417, 2.94909210675309568214675799174, 3.26205160426145124720208221811, 3.73736176759141151028476772946, 4.09656461779126087443721896265, 4.58830060096391877825894587037, 4.88095705277131245675559300256, 4.98774941147231156275822313531, 5.77147233262827214693079606726, 5.89686889359369810282120351182, 6.25515810304723313731381613557, 6.27618429580222546328848748775, 6.85098900121693001421745282720, 7.24077775462987778537210071402, 7.69851203475497045286173525712, 8.056280428413773612815340095489

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