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Properties

Label 4-2925e2-1.1-c1e2-0-15
Degree $4$
Conductor $8555625$
Sign $1$
Analytic cond. $545.514$
Root an. cond. $4.83282$
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  + 4·4-s + 12·11-s + 12·16-s + 8·19-s − 6·29-s − 8·31-s − 12·41-s + 48·44-s − 2·49-s − 12·59-s − 2·61-s + 32·64-s + 12·71-s + 32·76-s − 22·79-s + 30·101-s − 4·109-s − 24·116-s + 86·121-s − 32·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 2·4-s + 3.61·11-s + 3·16-s + 1.83·19-s − 1.11·29-s − 1.43·31-s − 1.87·41-s + 7.23·44-s − 2/7·49-s − 1.56·59-s − 0.256·61-s + 4·64-s + 1.42·71-s + 3.67·76-s − 2.47·79-s + 2.98·101-s − 0.383·109-s − 2.22·116-s + 7.81·121-s − 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8555625\)    =    \(3^{4} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(545.514\)
Root analytic conductor: \(4.83282\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8555625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.493452456\)
\(L(\frac12)\) \(\approx\) \(7.493452456\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
5 \( 1 \)
13$C_2$ \( 1 + T^{2} \)
good2$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.2.a_ae
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.11.am_cg
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.23.a_abl
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.29.g_cp
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.43.a_abl
47$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.47.a_adq
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.53.a_az
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.59.m_fy
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.61.c_et
67$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.67.a_ck
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.79.w_kt
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.83.a_afa
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.97.a_adq
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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

−8.983436881066984547369234034226, −8.582887181217430831922729378782, −8.205092011179646064713753335775, −7.43167987725047131298791554810, −7.28319117543009726833182006872, −7.23921661533166157057491223005, −6.58910835641676813175329582130, −6.43014381863079405019758494636, −6.05415906160622371351856363184, −5.67586533066318116765341780368, −5.26216606314948914912926332515, −4.64399664334264137383902787230, −3.96865150824403263756528565711, −3.67138392430991514183174852451, −3.23736977879255460318716532151, −3.14029777121873409736979204137, −2.09627615769300091486610904418, −1.66202816144099359772524993198, −1.49830945570426003814370886225, −0.905071541249066595623426828020, 0.905071541249066595623426828020, 1.49830945570426003814370886225, 1.66202816144099359772524993198, 2.09627615769300091486610904418, 3.14029777121873409736979204137, 3.23736977879255460318716532151, 3.67138392430991514183174852451, 3.96865150824403263756528565711, 4.64399664334264137383902787230, 5.26216606314948914912926332515, 5.67586533066318116765341780368, 6.05415906160622371351856363184, 6.43014381863079405019758494636, 6.58910835641676813175329582130, 7.23921661533166157057491223005, 7.28319117543009726833182006872, 7.43167987725047131298791554810, 8.205092011179646064713753335775, 8.582887181217430831922729378782, 8.983436881066984547369234034226

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