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Properties

Label 4-1352e2-1.1-c1e2-0-8
Degree $4$
Conductor $1827904$
Sign $1$
Analytic cond. $116.548$
Root an. cond. $3.28569$
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  − 3-s + 5-s − 5·7-s − 9-s − 2·11-s − 15-s + 3·17-s + 4·19-s + 5·21-s − 2·23-s − 5·25-s − 2·29-s − 2·31-s + 2·33-s − 5·35-s − 15·37-s − 16·41-s − 15·43-s − 45-s + 11·47-s + 9·49-s − 3·51-s + 8·53-s − 2·55-s − 4·57-s − 20·59-s + 14·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 1.88·7-s − 1/3·9-s − 0.603·11-s − 0.258·15-s + 0.727·17-s + 0.917·19-s + 1.09·21-s − 0.417·23-s − 25-s − 0.371·29-s − 0.359·31-s + 0.348·33-s − 0.845·35-s − 2.46·37-s − 2.49·41-s − 2.28·43-s − 0.149·45-s + 1.60·47-s + 9/7·49-s − 0.420·51-s + 1.09·53-s − 0.269·55-s − 0.529·57-s − 2.60·59-s + 1.79·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1827904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1827904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1827904\)    =    \(2^{6} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(116.548\)
Root analytic conductor: \(3.28569\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1827904,\ (\ :1/2, 1/2),\ 1)\)

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$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) 2.3.b_c
5$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_g
7$D_{4}$ \( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.7.f_q
11$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_g
17$D_{4}$ \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_bg
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_be
29$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.29.c_bq
31$D_{4}$ \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.31.c_bu
37$D_{4}$ \( 1 + 15 T + 126 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.37.p_ew
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.41.q_fq
43$D_{4}$ \( 1 + 15 T + 138 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.43.p_fi
47$D_{4}$ \( 1 - 11 T + 120 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.47.al_eq
53$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_cc
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.59.u_ik
61$D_{4}$ \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.61.ao_fy
67$D_{4}$ \( 1 + 10 T + 142 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.67.k_fm
71$D_{4}$ \( 1 - T + 104 T^{2} - p T^{3} + p^{2} T^{4} \) 2.71.ab_ea
73$D_{4}$ \( 1 + 6 T + 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.73.g_c
79$D_{4}$ \( 1 + 14 T + 190 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.79.o_hi
83$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \) 2.83.a_du
89$D_{4}$ \( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.89.c_gg
97$D_{4}$ \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_fa
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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

−9.595795679081987814530924330338, −9.054327494261541987045159112323, −8.626208959892337578155213827854, −8.315947493781645023799343228080, −7.69557165170746804820670841053, −7.12415532086964263361055871408, −7.02438097256606195514184623585, −6.48546828845935373415850598658, −6.01192676264472045502532537536, −5.66211951164486702805768400366, −5.32374685837033469222583888806, −5.05477510258922513636417059246, −4.13490036843377061867703674435, −3.60220851753558403078051680386, −3.17360351017457894252245244745, −2.98095114854347135050261094381, −1.99598667579343256841738245594, −1.48640200281167504830267848980, 0, 0, 1.48640200281167504830267848980, 1.99598667579343256841738245594, 2.98095114854347135050261094381, 3.17360351017457894252245244745, 3.60220851753558403078051680386, 4.13490036843377061867703674435, 5.05477510258922513636417059246, 5.32374685837033469222583888806, 5.66211951164486702805768400366, 6.01192676264472045502532537536, 6.48546828845935373415850598658, 7.02438097256606195514184623585, 7.12415532086964263361055871408, 7.69557165170746804820670841053, 8.315947493781645023799343228080, 8.626208959892337578155213827854, 9.054327494261541987045159112323, 9.595795679081987814530924330338

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