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Properties

Label 4-5440e2-1.1-c1e2-0-8
Degree $4$
Conductor $29593600$
Sign $1$
Analytic cond. $1886.91$
Root an. cond. $6.59079$
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 − 6·7-s + 2·11-s − 4·15-s − 2·17-s + 4·19-s − 12·21-s − 10·23-s + 3·25-s − 2·27-s + 2·31-s + 4·33-s + 12·35-s + 8·37-s − 4·43-s − 16·47-s + 16·49-s − 4·51-s + 4·53-s − 4·55-s + 8·57-s − 12·59-s + 12·61-s + 8·67-s − 20·69-s − 18·71-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s − 2.26·7-s + 0.603·11-s − 1.03·15-s − 0.485·17-s + 0.917·19-s − 2.61·21-s − 2.08·23-s + 3/5·25-s − 0.384·27-s + 0.359·31-s + 0.696·33-s + 2.02·35-s + 1.31·37-s − 0.609·43-s − 2.33·47-s + 16/7·49-s − 0.560·51-s + 0.549·53-s − 0.539·55-s + 1.05·57-s − 1.56·59-s + 1.53·61-s + 0.977·67-s − 2.40·69-s − 2.13·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29593600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29593600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(29593600\)    =    \(2^{12} \cdot 5^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1886.91\)
Root analytic conductor: \(6.59079\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 29593600,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
17$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_e
7$D_{4}$ \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.7.g_u
11$D_{4}$ \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_u
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
19$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_be
23$D_{4}$ \( 1 + 10 T + 68 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.23.k_cq
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.29.a_bu
31$D_{4}$ \( 1 - 2 T - 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.31.ac_am
37$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.37.ai_da
41$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.41.a_aba
43$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_bq
47$D_{4}$ \( 1 + 16 T + 146 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.47.q_fq
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.53.ae_eg
59$D_{4}$ \( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_fm
61$D_{4}$ \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.61.am_eg
67$D_{4}$ \( 1 - 8 T + 138 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_fi
71$D_{4}$ \( 1 + 18 T + 220 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.71.s_im
73$D_{4}$ \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_fu
79$D_{4}$ \( 1 + 2 T - 84 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_adg
83$D_{4}$ \( 1 - 4 T + 122 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_es
89$D_{4}$ \( 1 - 20 T + 266 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.89.au_kg
97$D_{4}$ \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.97.m_ha
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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

−7.928922963658874523758355003244, −7.80426719117419148135812987772, −7.35964288462788227602696607052, −6.83721870549610563750346602203, −6.56259374097146080810193118420, −6.40066565088391416726406346858, −5.87381654218097246350202106862, −5.69885573069703090362939070780, −4.91460372816291462828069273390, −4.59483975957788507831792685189, −3.99033601735098846326360927815, −3.82629336247194480501038923942, −3.33657356547385063368288699885, −3.21766165034034208013454350049, −2.68488056389833965110439082532, −2.43616370254769277937078573550, −1.70700465844956140791048896558, −1.02302862836564333269616234969, 0, 0, 1.02302862836564333269616234969, 1.70700465844956140791048896558, 2.43616370254769277937078573550, 2.68488056389833965110439082532, 3.21766165034034208013454350049, 3.33657356547385063368288699885, 3.82629336247194480501038923942, 3.99033601735098846326360927815, 4.59483975957788507831792685189, 4.91460372816291462828069273390, 5.69885573069703090362939070780, 5.87381654218097246350202106862, 6.40066565088391416726406346858, 6.56259374097146080810193118420, 6.83721870549610563750346602203, 7.35964288462788227602696607052, 7.80426719117419148135812987772, 7.928922963658874523758355003244

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