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Properties

Label 4-1960e2-1.1-c1e2-0-21
Degree $4$
Conductor $3841600$
Sign $1$
Analytic cond. $244.943$
Root an. cond. $3.95609$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·5-s − 9-s − 4·11-s − 4·13-s − 4·15-s + 4·17-s − 14·23-s + 3·25-s − 6·27-s − 10·29-s + 4·31-s − 8·33-s − 8·39-s − 6·41-s + 6·43-s + 2·45-s − 12·47-s + 8·51-s + 8·55-s + 8·59-s − 2·61-s + 8·65-s − 6·67-s − 28·69-s − 24·71-s + 4·73-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s − 1/3·9-s − 1.20·11-s − 1.10·13-s − 1.03·15-s + 0.970·17-s − 2.91·23-s + 3/5·25-s − 1.15·27-s − 1.85·29-s + 0.718·31-s − 1.39·33-s − 1.28·39-s − 0.937·41-s + 0.914·43-s + 0.298·45-s − 1.75·47-s + 1.12·51-s + 1.07·55-s + 1.04·59-s − 0.256·61-s + 0.992·65-s − 0.733·67-s − 3.37·69-s − 2.84·71-s + 0.468·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3841600\)    =    \(2^{6} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(244.943\)
Root analytic conductor: \(3.95609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3841600,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good3$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_f
11$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_s
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_g
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.19.a_g
23$D_{4}$ \( 1 + 14 T + 93 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.23.o_dp
29$D_{4}$ \( 1 + 10 T + 75 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.29.k_cx
31$D_{4}$ \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.31.ae_cg
37$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.37.a_bq
41$D_{4}$ \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_df
43$D_{4}$ \( 1 - 6 T - 3 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_ad
47$D_{4}$ \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_du
53$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.53.a_cw
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$D_{4}$ \( 1 + 2 T + 91 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_dn
67$D_{4}$ \( 1 + 6 T + 45 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.67.g_bt
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$D_{4}$ \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.73.ae_eo
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$D_{4}$ \( 1 - 18 T + 229 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.83.as_iv
89$D_{4}$ \( 1 - 22 T + 3 p T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.89.aw_kh
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.97.m_iw
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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

−8.756695762399085299275253108086, −8.619317018660247823745416537693, −8.060495892957879077127709056661, −7.79314431319499445826951175320, −7.58051557297763597212313521009, −7.52086004847455421744693340558, −6.60798012782247883720386075574, −6.26712608188395108020021832776, −5.57402492917759257637210691215, −5.50565187954816685475120150630, −4.86710536399246228016371647026, −4.41238527346293617996991422389, −3.77179814689222658238399266736, −3.59707277010430743651647565967, −3.01806601408796427452784010423, −2.53357739766442948989820418891, −2.21021868661513404253991244374, −1.51676762360876229218550540439, 0, 0, 1.51676762360876229218550540439, 2.21021868661513404253991244374, 2.53357739766442948989820418891, 3.01806601408796427452784010423, 3.59707277010430743651647565967, 3.77179814689222658238399266736, 4.41238527346293617996991422389, 4.86710536399246228016371647026, 5.50565187954816685475120150630, 5.57402492917759257637210691215, 6.26712608188395108020021832776, 6.60798012782247883720386075574, 7.52086004847455421744693340558, 7.58051557297763597212313521009, 7.79314431319499445826951175320, 8.060495892957879077127709056661, 8.619317018660247823745416537693, 8.756695762399085299275253108086

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