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Properties

Label 4-2376e2-1.1-c1e2-0-22
Degree $4$
Conductor $5645376$
Sign $1$
Analytic cond. $359.954$
Root an. cond. $4.35573$
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  − 2·7-s + 2·11-s − 8·13-s + 2·17-s + 4·19-s − 6·23-s − 2·25-s − 10·29-s − 4·31-s − 2·37-s − 6·41-s + 18·43-s − 10·47-s − 9·49-s − 8·53-s − 10·59-s + 12·67-s − 16·71-s − 16·73-s − 4·77-s − 2·79-s + 8·83-s + 4·89-s + 16·91-s − 6·97-s − 14·101-s − 16·107-s + ⋯
L(s)  = 1  − 0.755·7-s + 0.603·11-s − 2.21·13-s + 0.485·17-s + 0.917·19-s − 1.25·23-s − 2/5·25-s − 1.85·29-s − 0.718·31-s − 0.328·37-s − 0.937·41-s + 2.74·43-s − 1.45·47-s − 9/7·49-s − 1.09·53-s − 1.30·59-s + 1.46·67-s − 1.89·71-s − 1.87·73-s − 0.455·77-s − 0.225·79-s + 0.878·83-s + 0.423·89-s + 1.67·91-s − 0.609·97-s − 1.39·101-s − 1.54·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5645376\)    =    \(2^{6} \cdot 3^{6} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(359.954\)
Root analytic conductor: \(4.35573\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5645376,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$D_{4}$ \( 1 + 2 T + 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_n
13$D_{4}$ \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.13.i_bi
17$D_{4}$ \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_r
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$D_{4}$ \( 1 + 6 T + p T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_x
29$D_{4}$ \( 1 + 10 T + 81 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.29.k_dd
31$D_{4}$ \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.31.e_bi
37$D_{4}$ \( 1 + 2 T + 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_br
41$D_{4}$ \( 1 + 6 T + 89 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_dl
43$D_{4}$ \( 1 - 18 T + 165 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.43.as_gj
47$D_{4}$ \( 1 + 10 T + p T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.47.k_bv
53$D_{4}$ \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.53.i_dm
59$D_{4}$ \( 1 + 10 T + 135 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.59.k_ff
61$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \) 2.61.a_ek
67$D_{4}$ \( 1 - 12 T + 138 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.67.am_fi
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$D_{4}$ \( 1 + 16 T + 178 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.73.q_gw
79$D_{4}$ \( 1 + 2 T + 157 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_gb
83$D_{4}$ \( 1 - 8 T + 174 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.83.ai_gs
89$D_{4}$ \( 1 - 4 T + 110 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.89.ae_eg
97$D_{4}$ \( 1 + 6 T + 3 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.97.g_d
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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

−9.108992793179020898863860259177, −8.226785871676914905880499902688, −7.946810446432417407172800797989, −7.56528745192439806594683194523, −7.27193620935984775545197298012, −7.04353667121140534427886093013, −6.26337758419114815063522016948, −6.23854164085668318485501224959, −5.47479729199168279571050096180, −5.41659851869620813061707805394, −4.78085088315414941298708677777, −4.39459800551560453603864013118, −3.77394207820132829799787890118, −3.54428295501618927842234527994, −2.87732194832855383623921284403, −2.56047259694375897315656576168, −1.78882936347537665140239729312, −1.44521518068066023617082503862, 0, 0, 1.44521518068066023617082503862, 1.78882936347537665140239729312, 2.56047259694375897315656576168, 2.87732194832855383623921284403, 3.54428295501618927842234527994, 3.77394207820132829799787890118, 4.39459800551560453603864013118, 4.78085088315414941298708677777, 5.41659851869620813061707805394, 5.47479729199168279571050096180, 6.23854164085668318485501224959, 6.26337758419114815063522016948, 7.04353667121140534427886093013, 7.27193620935984775545197298012, 7.56528745192439806594683194523, 7.946810446432417407172800797989, 8.226785871676914905880499902688, 9.108992793179020898863860259177

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