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Properties

Label 4-1140e2-1.1-c1e2-0-11
Degree $4$
Conductor $1299600$
Sign $-1$
Analytic cond. $82.8636$
Root an. cond. $3.01710$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 2·5-s + 3·8-s + 9-s + 2·10-s − 16-s − 12·17-s − 18-s + 2·20-s + 3·25-s + 8·29-s − 5·32-s + 12·34-s − 36-s + 8·37-s − 6·40-s − 2·45-s − 10·49-s − 3·50-s + 4·53-s − 8·58-s + 4·61-s + 7·64-s + 12·68-s + 3·72-s + 28·73-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.894·5-s + 1.06·8-s + 1/3·9-s + 0.632·10-s − 1/4·16-s − 2.91·17-s − 0.235·18-s + 0.447·20-s + 3/5·25-s + 1.48·29-s − 0.883·32-s + 2.05·34-s − 1/6·36-s + 1.31·37-s − 0.948·40-s − 0.298·45-s − 1.42·49-s − 0.424·50-s + 0.549·53-s − 1.05·58-s + 0.512·61-s + 7/8·64-s + 1.45·68-s + 0.353·72-s + 3.27·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1299600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1299600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1299600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(82.8636\)
Root analytic conductor: \(3.01710\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1299600,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T + p T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$ \( ( 1 + T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.29.ai_cw
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.37.ai_dm
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.43.a_de
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.53.ae_eg
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.a_cs
71$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.a_aek
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.73.abc_ne
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.97.y_na
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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

−7.944956574862892187546933534941, −7.46307212019793678556314757234, −6.76242068169946844350861892331, −6.69343741358824607018216997419, −6.35568709931231127456553997796, −5.30054231386447702035951774158, −5.08269165937541766478904797647, −4.42342128541001589497390333987, −4.15312467491244982662604138290, −3.93065235513561826916389859535, −2.92380519859586694297776399100, −2.45875404104195454497297562929, −1.70399671261848221618934256367, −0.818960836178036513910564281349, 0, 0.818960836178036513910564281349, 1.70399671261848221618934256367, 2.45875404104195454497297562929, 2.92380519859586694297776399100, 3.93065235513561826916389859535, 4.15312467491244982662604138290, 4.42342128541001589497390333987, 5.08269165937541766478904797647, 5.30054231386447702035951774158, 6.35568709931231127456553997796, 6.69343741358824607018216997419, 6.76242068169946844350861892331, 7.46307212019793678556314757234, 7.944956574862892187546933534941

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