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Properties

Label 4-5440e2-1.1-c1e2-0-9
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 2·5-s + 4·7-s + 8·9-s − 8·11-s − 8·15-s − 2·17-s − 16·21-s + 4·23-s + 3·25-s − 12·27-s + 4·29-s + 32·33-s + 8·35-s + 4·37-s + 4·41-s + 4·43-s + 16·45-s + 4·47-s + 8·51-s − 12·53-s − 16·55-s − 24·59-s − 4·61-s + 32·63-s − 12·67-s − 16·69-s + ⋯
L(s)  = 1  − 2.30·3-s + 0.894·5-s + 1.51·7-s + 8/3·9-s − 2.41·11-s − 2.06·15-s − 0.485·17-s − 3.49·21-s + 0.834·23-s + 3/5·25-s − 2.30·27-s + 0.742·29-s + 5.57·33-s + 1.35·35-s + 0.657·37-s + 0.624·41-s + 0.609·43-s + 2.38·45-s + 0.583·47-s + 1.12·51-s − 1.64·53-s − 2.15·55-s − 3.12·59-s − 0.512·61-s + 4.03·63-s − 1.46·67-s − 1.92·69-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$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.3.e_i
7$D_{4}$ \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_q
11$D_{4}$ \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.11.i_bk
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.19.a_be
23$D_{4}$ \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_bw
29$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_cc
31$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.31.a_bs
37$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_g
41$D_{4}$ \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.41.ae_o
43$D_{4}$ \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_cg
47$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_dm
53$D_{4}$ \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.53.m_eg
59$D_{4}$ \( 1 + 24 T + 254 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.59.y_ju
61$D_{4}$ \( 1 + 4 T + 94 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_dq
67$D_{4}$ \( 1 + 12 T + 162 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.67.m_gg
71$C_2^2$ \( 1 + 124 T^{2} + p^{2} T^{4} \) 2.71.a_eu
73$D_{4}$ \( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.73.e_fm
79$D_{4}$ \( 1 + 8 T + 172 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_gq
83$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_bq
89$D_{4}$ \( 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.89.q_ic
97$D_{4}$ \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_gk
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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

−7.903572996808499233526008689882, −7.52302911297560750684352264914, −7.29095278690640120189683231487, −6.91218574126440561547091401266, −6.17409497391199646221971748727, −6.13446226872738893336959817558, −5.76616025170242994538531998760, −5.60166099673930416174245469860, −4.92960934004069159478998374398, −4.90522896341215232567281129429, −4.55366942509113944682210834767, −4.50345307445020084407369677475, −3.42826872851663667752452248076, −2.87517576236151676139880390216, −2.53923013459639099598653057876, −2.01791843607347906683070829071, −1.34928197971440321054106538568, −1.13690018436444611787123090264, 0, 0, 1.13690018436444611787123090264, 1.34928197971440321054106538568, 2.01791843607347906683070829071, 2.53923013459639099598653057876, 2.87517576236151676139880390216, 3.42826872851663667752452248076, 4.50345307445020084407369677475, 4.55366942509113944682210834767, 4.90522896341215232567281129429, 4.92960934004069159478998374398, 5.60166099673930416174245469860, 5.76616025170242994538531998760, 6.13446226872738893336959817558, 6.17409497391199646221971748727, 6.91218574126440561547091401266, 7.29095278690640120189683231487, 7.52302911297560750684352264914, 7.903572996808499233526008689882

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