Properties

Label 4-2299e2-1.1-c1e2-0-4
Degree $4$
Conductor $5285401$
Sign $1$
Analytic cond. $337.001$
Root an. cond. $4.28457$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 6·5-s + 2·7-s + 4·8-s − 3·9-s + 12·10-s + 2·13-s + 4·14-s + 8·16-s + 6·17-s − 6·18-s + 2·19-s + 12·20-s + 6·23-s + 17·25-s + 4·26-s + 4·28-s + 14·29-s − 16·31-s + 8·32-s + 12·34-s + 12·35-s − 6·36-s + 8·37-s + 4·38-s + 24·40-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 2.68·5-s + 0.755·7-s + 1.41·8-s − 9-s + 3.79·10-s + 0.554·13-s + 1.06·14-s + 2·16-s + 1.45·17-s − 1.41·18-s + 0.458·19-s + 2.68·20-s + 1.25·23-s + 17/5·25-s + 0.784·26-s + 0.755·28-s + 2.59·29-s − 2.87·31-s + 1.41·32-s + 2.05·34-s + 2.02·35-s − 36-s + 1.31·37-s + 0.648·38-s + 3.79·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5285401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5285401 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5285401\)    =    \(11^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(337.001\)
Root analytic conductor: \(4.28457\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5285401,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(14.32852063\)
\(L(\frac12)\) \(\approx\) \(14.32852063\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad11 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.2.ac_c
3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.3.a_d
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.5.ag_t
7$D_{4}$ \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_m
13$D_{4}$ \( 1 - 2 T - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_a
17$D_{4}$ \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_bo
23$D_{4}$ \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_br
29$D_{4}$ \( 1 - 14 T + 104 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.29.ao_ea
31$D_{4}$ \( 1 + 16 T + 123 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.31.q_et
37$D_{4}$ \( 1 - 8 T + 87 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.37.ai_dj
41$D_{4}$ \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.41.ai_by
43$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_cc
47$D_{4}$ \( 1 + 16 T + 146 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.47.q_fq
53$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.53.a_cg
59$D_{4}$ \( 1 - 4 T + 47 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.59.ae_bv
61$D_{4}$ \( 1 - 2 T + 48 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_bw
67$D_{4}$ \( 1 - 4 T + 63 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_cl
71$D_{4}$ \( 1 + 20 T + 239 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.71.u_jf
73$D_{4}$ \( 1 - 20 T + 234 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.73.au_ja
79$D_{4}$ \( 1 + 14 T + 204 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.79.o_hw
83$D_{4}$ \( 1 + 18 T + 220 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.83.s_im
89$D_{4}$ \( 1 - 4 T + 155 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.89.ae_fz
97$D_{4}$ \( 1 + 12 T + 203 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.97.m_hv
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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

−9.345474458783514479127475617929, −8.805778814879333476428498898046, −8.426736306701090200128870971072, −7.984911504181618517681947248070, −7.72053775463671693124271606445, −6.95226643755423387034911518671, −6.76400400392052759642681681830, −6.25774186943616057118958119049, −5.75015615803362537751485084773, −5.65239629362665155051279818597, −5.21295032461962545099916472989, −5.13197214691892329488244702683, −4.55859564085367277148380255121, −4.06768569613896955940281053284, −3.18997221927042840363442727300, −3.12867828523185399923929214084, −2.57323260781065661654825136480, −1.89574217642569941110524695291, −1.37754479462097346506318181566, −1.24740068046015055985093274803, 1.24740068046015055985093274803, 1.37754479462097346506318181566, 1.89574217642569941110524695291, 2.57323260781065661654825136480, 3.12867828523185399923929214084, 3.18997221927042840363442727300, 4.06768569613896955940281053284, 4.55859564085367277148380255121, 5.13197214691892329488244702683, 5.21295032461962545099916472989, 5.65239629362665155051279818597, 5.75015615803362537751485084773, 6.25774186943616057118958119049, 6.76400400392052759642681681830, 6.95226643755423387034911518671, 7.72053775463671693124271606445, 7.984911504181618517681947248070, 8.426736306701090200128870971072, 8.805778814879333476428498898046, 9.345474458783514479127475617929

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