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Properties

Label 4-930e2-1.1-c1e2-0-15
Degree $4$
Conductor $864900$
Sign $1$
Analytic cond. $55.1467$
Root an. cond. $2.72508$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 3·4-s − 5-s − 2·6-s + 4·8-s − 2·10-s + 11-s − 3·12-s + 6·13-s + 15-s + 5·16-s + 17-s − 4·19-s − 3·20-s + 2·22-s + 14·23-s − 4·24-s + 12·26-s + 27-s + 12·29-s + 2·30-s + 4·31-s + 6·32-s − 33-s + 2·34-s − 7·37-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.447·5-s − 0.816·6-s + 1.41·8-s − 0.632·10-s + 0.301·11-s − 0.866·12-s + 1.66·13-s + 0.258·15-s + 5/4·16-s + 0.242·17-s − 0.917·19-s − 0.670·20-s + 0.426·22-s + 2.91·23-s − 0.816·24-s + 2.35·26-s + 0.192·27-s + 2.22·29-s + 0.365·30-s + 0.718·31-s + 1.06·32-s − 0.174·33-s + 0.342·34-s − 1.15·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(864900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(55.1467\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 864900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.905039725\)
\(L(\frac12)\) \(\approx\) \(4.905039725\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3$C_2$ \( 1 + T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
31$C_2$ \( 1 - 4 T + p T^{2} \)
good7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.7.a_ah
11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) 2.11.ab_ak
13$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_x
17$C_2^2$ \( 1 - T - 16 T^{2} - p T^{3} + p^{2} T^{4} \) 2.17.ab_aq
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_ad
23$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.23.ao_dr
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
37$C_2^2$ \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.37.h_m
41$C_2^2$ \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.41.ac_abl
43$C_2^2$ \( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.43.ah_g
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.47.ag_dz
53$C_2^2$ \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_abl
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_dh
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.61.bc_mg
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.f_abq
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_abj
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_acr
79$C_2^2$ \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.79.d_acs
83$C_2^2$ \( 1 + 16 T + 173 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.83.q_gr
89$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.89.ai_hm
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.97.m_iw
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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

−10.45078034470625065355973118830, −10.38358508955195293006385249304, −9.265983887851629742557743782328, −8.938021436634257134779824726231, −8.681136048549815276933184667397, −8.151449732357854426048694566240, −7.48284618114923384848553218494, −7.20575726398531714710647108545, −6.52919672020546172684816452615, −6.47854961586060048717879920478, −5.80717677167120521784200319955, −5.65169968243665552542141871900, −4.75764642334540718625258009648, −4.59919520368904898218184099730, −4.21290607583222055117696661273, −3.46724156998708297664841128708, −3.00785757185597485027405220046, −2.68314329674674070251577371413, −1.44378102724780495009563903983, −0.998065024629973811541892675766, 0.998065024629973811541892675766, 1.44378102724780495009563903983, 2.68314329674674070251577371413, 3.00785757185597485027405220046, 3.46724156998708297664841128708, 4.21290607583222055117696661273, 4.59919520368904898218184099730, 4.75764642334540718625258009648, 5.65169968243665552542141871900, 5.80717677167120521784200319955, 6.47854961586060048717879920478, 6.52919672020546172684816452615, 7.20575726398531714710647108545, 7.48284618114923384848553218494, 8.151449732357854426048694566240, 8.681136048549815276933184667397, 8.938021436634257134779824726231, 9.265983887851629742557743782328, 10.38358508955195293006385249304, 10.45078034470625065355973118830

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