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Properties

Label 4-648675-1.1-c1e2-0-0
Degree $4$
Conductor $648675$
Sign $-1$
Analytic cond. $41.3600$
Root an. cond. $2.53597$
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  + 3-s − 3·4-s − 8·7-s + 9-s − 3·12-s + 4·13-s + 5·16-s − 8·19-s − 8·21-s + 25-s + 27-s + 24·28-s − 2·31-s − 3·36-s + 20·37-s + 4·39-s − 24·43-s + 5·48-s + 34·49-s − 12·52-s − 8·57-s + 12·61-s − 8·63-s − 3·64-s + 16·67-s + 12·73-s + 75-s + ⋯
L(s)  = 1  + 0.577·3-s − 3/2·4-s − 3.02·7-s + 1/3·9-s − 0.866·12-s + 1.10·13-s + 5/4·16-s − 1.83·19-s − 1.74·21-s + 1/5·25-s + 0.192·27-s + 4.53·28-s − 0.359·31-s − 1/2·36-s + 3.28·37-s + 0.640·39-s − 3.65·43-s + 0.721·48-s + 34/7·49-s − 1.66·52-s − 1.05·57-s + 1.53·61-s − 1.00·63-s − 3/8·64-s + 1.95·67-s + 1.40·73-s + 0.115·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(648675\)    =    \(3^{3} \cdot 5^{2} \cdot 31^{2}\)
Sign: $-1$
Analytic conductor: \(41.3600\)
Root analytic conductor: \(2.53597\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 648675,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad3$C_1$ \( 1 - T \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
31$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.2.a_d
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.7.i_be
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.37.au_gs
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.43.y_iw
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.a_dy
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.a_cc
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
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

−8.161265561981018255582993597565, −8.084796703174245407898935126609, −7.02137188315382950394322884244, −6.72985993744660244263986196267, −6.31086051986075869789608308348, −6.06865597150852415394362452213, −5.42330275236629278699125705447, −4.69288327276767791313809475264, −4.18102881376974849679118546204, −3.68795247182319804330464993604, −3.47701333779520587015827014256, −2.87846544760766985461384480320, −2.17254925636267642483655615973, −0.812096310306728023697107886848, 0, 0.812096310306728023697107886848, 2.17254925636267642483655615973, 2.87846544760766985461384480320, 3.47701333779520587015827014256, 3.68795247182319804330464993604, 4.18102881376974849679118546204, 4.69288327276767791313809475264, 5.42330275236629278699125705447, 6.06865597150852415394362452213, 6.31086051986075869789608308348, 6.72985993744660244263986196267, 7.02137188315382950394322884244, 8.084796703174245407898935126609, 8.161265561981018255582993597565

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