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Properties

Label 4-5440e2-1.1-c1e2-0-1
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·5-s + 2·7-s + 6·11-s + 8·13-s − 4·15-s − 2·17-s + 4·19-s + 4·21-s + 6·23-s + 3·25-s − 2·27-s − 10·31-s + 12·33-s − 4·35-s + 8·37-s + 16·39-s − 8·43-s − 12·47-s − 8·49-s − 4·51-s − 12·53-s − 12·55-s + 8·57-s + 12·59-s − 4·61-s − 16·65-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 0.755·7-s + 1.80·11-s + 2.21·13-s − 1.03·15-s − 0.485·17-s + 0.917·19-s + 0.872·21-s + 1.25·23-s + 3/5·25-s − 0.384·27-s − 1.79·31-s + 2.08·33-s − 0.676·35-s + 1.31·37-s + 2.56·39-s − 1.21·43-s − 1.75·47-s − 8/7·49-s − 0.560·51-s − 1.64·53-s − 1.61·55-s + 1.05·57-s + 1.56·59-s − 0.512·61-s − 1.98·65-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: \(0\)
Selberg data: \((4,\ 29593600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.501645427\)
\(L(\frac12)\) \(\approx\) \(5.501645427\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_m
11$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_bc
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
19$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_be
23$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_bc
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.29.a_bu
31$D_{4}$ \( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_dg
37$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.37.ai_da
41$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.41.a_cs
43$D_{4}$ \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_dm
47$D_{4}$ \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_de
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$D_{4}$ \( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_fm
61$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_da
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.67.u_ja
71$D_{4}$ \( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_cy
73$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_cc
79$D_{4}$ \( 1 - 2 T - 84 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.79.ac_adg
83$D_{4}$ \( 1 - 24 T + 298 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.83.ay_lm
89$D_{4}$ \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_ec
97$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.97.ae_fu
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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

−8.418932434875304339383533902028, −8.091947923321874936944318706906, −7.62172547158191915565848376455, −7.53022300481004052980919832815, −6.93158190902663312297991196788, −6.60356084493760610975077921788, −6.24604990357539231366525336009, −6.01414007510040607797489850788, −5.41272267118561332463103544981, −4.95586032889971216760528060251, −4.49878910362399220280749078835, −4.31502102771274178378664580755, −3.57864563069013220333310165727, −3.57029175687943243123822306482, −3.07413615138365855489493762842, −3.01155377904767149931182389625, −1.79108898698802581369322659185, −1.75625622697916060009938648346, −1.24126190983024370856865880454, −0.60564231137073375066833627473, 0.60564231137073375066833627473, 1.24126190983024370856865880454, 1.75625622697916060009938648346, 1.79108898698802581369322659185, 3.01155377904767149931182389625, 3.07413615138365855489493762842, 3.57029175687943243123822306482, 3.57864563069013220333310165727, 4.31502102771274178378664580755, 4.49878910362399220280749078835, 4.95586032889971216760528060251, 5.41272267118561332463103544981, 6.01414007510040607797489850788, 6.24604990357539231366525336009, 6.60356084493760610975077921788, 6.93158190902663312297991196788, 7.53022300481004052980919832815, 7.62172547158191915565848376455, 8.091947923321874936944318706906, 8.418932434875304339383533902028

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