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Properties

Label 4-1632e2-1.1-c1e2-0-57
Degree $4$
Conductor $2663424$
Sign $1$
Analytic cond. $169.822$
Root an. cond. $3.60992$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s − 2·17-s − 8·19-s − 10·25-s − 4·27-s − 20·41-s − 8·43-s − 10·49-s + 4·51-s + 16·57-s + 8·59-s − 8·67-s − 28·73-s + 20·75-s + 5·81-s + 24·83-s − 4·89-s − 4·97-s + 32·107-s − 12·113-s − 22·121-s + 40·123-s + 127-s + 16·129-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s − 0.485·17-s − 1.83·19-s − 2·25-s − 0.769·27-s − 3.12·41-s − 1.21·43-s − 1.42·49-s + 0.560·51-s + 2.11·57-s + 1.04·59-s − 0.977·67-s − 3.27·73-s + 2.30·75-s + 5/9·81-s + 2.63·83-s − 0.423·89-s − 0.406·97-s + 3.09·107-s − 1.12·113-s − 2·121-s + 3.60·123-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2663424\)    =    \(2^{10} \cdot 3^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(169.822\)
Root analytic conductor: \(3.60992\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2663424,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
17$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.23.a_bq
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.a_ba
37$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.37.a_cw
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.41.u_ha
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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.71.a_fi
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.73.bc_ne
79$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.79.a_es
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.83.ay_ly
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.89.e_ha
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−7.03118715673001635895164549850, −6.69989082812922800047354757550, −6.29392484047154187578719849324, −6.06188611910220482610867547311, −5.53399853944419593600376639581, −5.04753779352725801743199793018, −4.63834490626473839787470699777, −4.32146836767987654020354473360, −3.63470410211943613588864920865, −3.38688660560456486653785466782, −2.39845581420990817522811157438, −1.86322179819912387769835446845, −1.46813220159740528632510266297, 0, 0, 1.46813220159740528632510266297, 1.86322179819912387769835446845, 2.39845581420990817522811157438, 3.38688660560456486653785466782, 3.63470410211943613588864920865, 4.32146836767987654020354473360, 4.63834490626473839787470699777, 5.04753779352725801743199793018, 5.53399853944419593600376639581, 6.06188611910220482610867547311, 6.29392484047154187578719849324, 6.69989082812922800047354757550, 7.03118715673001635895164549850

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