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Properties

Label 4-5408e2-1.1-c1e2-0-13
Degree $4$
Conductor $29246464$
Sign $1$
Analytic cond. $1864.77$
Root an. cond. $6.57138$
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  + 2·3-s − 6·7-s − 9-s + 2·11-s − 6·17-s − 6·19-s − 12·21-s + 6·23-s − 2·25-s − 6·27-s + 6·29-s + 4·33-s + 2·37-s − 6·41-s + 6·43-s − 12·47-s + 15·49-s − 12·51-s − 12·57-s − 6·59-s + 14·61-s + 6·63-s − 18·67-s + 12·69-s + 6·71-s + 8·73-s − 4·75-s + ⋯
L(s)  = 1  + 1.15·3-s − 2.26·7-s − 1/3·9-s + 0.603·11-s − 1.45·17-s − 1.37·19-s − 2.61·21-s + 1.25·23-s − 2/5·25-s − 1.15·27-s + 1.11·29-s + 0.696·33-s + 0.328·37-s − 0.937·41-s + 0.914·43-s − 1.75·47-s + 15/7·49-s − 1.68·51-s − 1.58·57-s − 0.781·59-s + 1.79·61-s + 0.755·63-s − 2.19·67-s + 1.44·69-s + 0.712·71-s + 0.936·73-s − 0.461·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29246464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29246464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(29246464\)    =    \(2^{10} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1864.77\)
Root analytic conductor: \(6.57138\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 29246464,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
13 \( 1 \)
good3$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_f
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$D_{4}$ \( 1 + 6 T + 3 p T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.7.g_v
11$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_f
17$D_{4}$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_bj
19$D_{4}$ \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_bd
23$D_{4}$ \( 1 - 6 T + 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_bl
29$D_{4}$ \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_bj
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.31.a_be
37$D_{4}$ \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_d
41$D_{4}$ \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_df
43$D_{4}$ \( 1 - 6 T + 45 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_bt
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.47.m_fa
53$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \) 2.53.a_du
59$D_{4}$ \( 1 + 6 T + 109 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_ef
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.61.ao_gp
67$D_{4}$ \( 1 + 18 T + 197 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.67.s_hp
71$D_{4}$ \( 1 - 6 T + 133 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_fd
73$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_dm
79$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.79.am_hm
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.83.i_ha
89$D_{4}$ \( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.89.s_it
97$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.97.s_kp
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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

−8.017101446729315471401142100435, −7.87817226205003433952809609159, −6.97801848379506627827446212801, −6.88862185089228911593369873000, −6.62594493305869202975504281596, −6.43930058916667015176272063507, −5.85871496758198323548148601866, −5.72034789108625399316460283236, −4.93432526474318105943096375768, −4.63169288849821074974347051267, −4.13896990221499182358350619544, −3.76791580286838541430769164191, −3.36404571879753064701560117660, −3.06122981171608459847492081771, −2.56571046910205261277043273995, −2.49754545051375097154283043208, −1.80268290889182784593386835245, −1.06708543261889983059449495965, 0, 0, 1.06708543261889983059449495965, 1.80268290889182784593386835245, 2.49754545051375097154283043208, 2.56571046910205261277043273995, 3.06122981171608459847492081771, 3.36404571879753064701560117660, 3.76791580286838541430769164191, 4.13896990221499182358350619544, 4.63169288849821074974347051267, 4.93432526474318105943096375768, 5.72034789108625399316460283236, 5.85871496758198323548148601866, 6.43930058916667015176272063507, 6.62594493305869202975504281596, 6.88862185089228911593369873000, 6.97801848379506627827446212801, 7.87817226205003433952809609159, 8.017101446729315471401142100435

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