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Properties

Label 4-1248e2-1.1-c1e2-0-11
Degree $4$
Conductor $1557504$
Sign $-1$
Analytic cond. $99.3078$
Root an. cond. $3.15679$
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  − 2·3-s + 3·9-s − 12·11-s + 4·17-s − 10·25-s − 4·27-s + 24·33-s − 8·43-s − 14·49-s − 8·51-s + 12·59-s + 24·67-s + 12·73-s + 20·75-s + 5·81-s − 12·83-s + 8·89-s + 28·97-s − 36·99-s + 16·107-s − 20·113-s + 86·121-s + 127-s + 16·129-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s − 3.61·11-s + 0.970·17-s − 2·25-s − 0.769·27-s + 4.17·33-s − 1.21·43-s − 2·49-s − 1.12·51-s + 1.56·59-s + 2.93·67-s + 1.40·73-s + 2.30·75-s + 5/9·81-s − 1.31·83-s + 0.847·89-s + 2.84·97-s − 3.61·99-s + 1.54·107-s − 1.88·113-s + 7.81·121-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1557504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1557504 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1557504\)    =    \(2^{10} \cdot 3^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(99.3078\)
Root analytic conductor: \(3.15679\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1557504,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.11.m_cg
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.47.a_ag
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.a_g
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.a_di
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.71.a_fi
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.a_adu
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.83.m_hu
89$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.89.ai_hm
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.97.abc_pa
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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

−7.70721800445715600502657637543, −7.42908510259424397801194980381, −6.61084979929999274945585321507, −6.46852629489327197023949075773, −5.56597126052820488051950415035, −5.50835748283028813340348869563, −5.25010915829523959096570683485, −4.80983646216625589523400190905, −4.21259172810034728687825987566, −3.41767268718365275694731336535, −3.12961052873891392238054519582, −2.16189239262552998925883882829, −2.05225824663803178760552738615, −0.69386451230752267708646070580, 0, 0.69386451230752267708646070580, 2.05225824663803178760552738615, 2.16189239262552998925883882829, 3.12961052873891392238054519582, 3.41767268718365275694731336535, 4.21259172810034728687825987566, 4.80983646216625589523400190905, 5.25010915829523959096570683485, 5.50835748283028813340348869563, 5.56597126052820488051950415035, 6.46852629489327197023949075773, 6.61084979929999274945585321507, 7.42908510259424397801194980381, 7.70721800445715600502657637543

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