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Properties

Label 4-1176e2-1.1-c1e2-0-49
Degree $4$
Conductor $1382976$
Sign $-1$
Analytic cond. $88.1797$
Root an. cond. $3.06437$
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  + 9-s + 8·23-s − 10·25-s + 4·29-s − 12·37-s + 8·43-s + 12·53-s − 8·67-s − 24·71-s − 32·79-s + 81-s + 16·107-s − 28·109-s + 4·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 1/3·9-s + 1.66·23-s − 2·25-s + 0.742·29-s − 1.97·37-s + 1.21·43-s + 1.64·53-s − 0.977·67-s − 2.84·71-s − 3.60·79-s + 1/9·81-s + 1.54·107-s − 2.68·109-s + 0.376·113-s − 2·121-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 + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1382976\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{4}\)
Sign: $-1$
Analytic conductor: \(88.1797\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1382976,\ (\ :1/2, 1/2),\ -1)\)

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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7 \( 1 \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.23.ai_ck
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.a_ack
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.61.a_ec
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.73.a_de
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.79.bg_py
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.89.a_gg
97$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.97.a_ack
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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

−7.41005169816589214200697437145, −7.39368277172255819158994073993, −7.06575029892920470972632576297, −6.43859371104521209533782451242, −5.87340118343345405222811842346, −5.62427172231298026532892197535, −5.13242773052280229930287694957, −4.40915109577399770914652954180, −4.26842049846374598919859794951, −3.57155516860156643593206388626, −3.02749641502338293168200065073, −2.52790453796007379380617788419, −1.73728899486802098617607613127, −1.17836444974596423542040449778, 0, 1.17836444974596423542040449778, 1.73728899486802098617607613127, 2.52790453796007379380617788419, 3.02749641502338293168200065073, 3.57155516860156643593206388626, 4.26842049846374598919859794951, 4.40915109577399770914652954180, 5.13242773052280229930287694957, 5.62427172231298026532892197535, 5.87340118343345405222811842346, 6.43859371104521209533782451242, 7.06575029892920470972632576297, 7.39368277172255819158994073993, 7.41005169816589214200697437145

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