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Properties

Label 4-31212-1.1-c1e2-0-3
Degree $4$
Conductor $31212$
Sign $-1$
Analytic cond. $1.99010$
Root an. cond. $1.18773$
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 + 4-s − 4·7-s + 9-s − 12-s − 12·13-s + 16-s + 8·19-s + 4·21-s + 6·25-s − 27-s − 4·28-s − 12·31-s + 36-s − 8·37-s + 12·39-s − 8·43-s − 48-s − 2·49-s − 12·52-s − 8·57-s − 8·61-s − 4·63-s + 64-s − 24·67-s + 4·73-s − 6·75-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s − 1.51·7-s + 1/3·9-s − 0.288·12-s − 3.32·13-s + 1/4·16-s + 1.83·19-s + 0.872·21-s + 6/5·25-s − 0.192·27-s − 0.755·28-s − 2.15·31-s + 1/6·36-s − 1.31·37-s + 1.92·39-s − 1.21·43-s − 0.144·48-s − 2/7·49-s − 1.66·52-s − 1.05·57-s − 1.02·61-s − 0.503·63-s + 1/8·64-s − 2.93·67-s + 0.468·73-s − 0.692·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31212 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31212 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(31212\)    =    \(2^{2} \cdot 3^{3} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(1.99010\)
Root analytic conductor: \(1.18773\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 31212,\ (\ :1/2, 1/2),\ -1)\)

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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 + T \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.13.m_ck
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.a_bq
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.31.m_du
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.a_da
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.a_dy
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.61.i_fi
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.79.au_jy
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.89.a_gs
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.97.am_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

−10.13636675520507553497406453887, −9.809680223299837765262582433989, −9.330186221363971309576111465163, −8.951168622566808298087877447389, −7.58460004774890445683046626123, −7.50224155158573118874050462938, −6.99640780173569532110131873676, −6.52498883063792512928187502617, −5.71929507199710618712172159386, −5.00716852164944696644803170219, −4.84625070630626219217033988858, −3.34589616522309311397390467846, −3.08824258991321588799800323044, −2.01050304782327789083256558357, 0, 2.01050304782327789083256558357, 3.08824258991321588799800323044, 3.34589616522309311397390467846, 4.84625070630626219217033988858, 5.00716852164944696644803170219, 5.71929507199710618712172159386, 6.52498883063792512928187502617, 6.99640780173569532110131873676, 7.50224155158573118874050462938, 7.58460004774890445683046626123, 8.951168622566808298087877447389, 9.330186221363971309576111465163, 9.809680223299837765262582433989, 10.13636675520507553497406453887

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