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Properties

Label 4-1782272-1.1-c1e2-0-3
Degree $4$
Conductor $1782272$
Sign $-1$
Analytic cond. $113.639$
Root an. cond. $3.26499$
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  + 4·3-s + 6·9-s + 2·11-s − 2·17-s − 6·25-s − 4·27-s + 8·33-s − 6·41-s − 2·43-s − 13·49-s − 8·51-s + 2·59-s + 8·67-s − 12·73-s − 24·75-s − 37·81-s − 6·83-s + 4·89-s − 20·97-s + 12·99-s − 12·107-s − 8·113-s − 19·121-s − 24·123-s + 127-s − 8·129-s + 131-s + ⋯
L(s)  = 1  + 2.30·3-s + 2·9-s + 0.603·11-s − 0.485·17-s − 6/5·25-s − 0.769·27-s + 1.39·33-s − 0.937·41-s − 0.304·43-s − 1.85·49-s − 1.12·51-s + 0.260·59-s + 0.977·67-s − 1.40·73-s − 2.77·75-s − 4.11·81-s − 0.658·83-s + 0.423·89-s − 2.03·97-s + 1.20·99-s − 1.16·107-s − 0.752·113-s − 1.72·121-s − 2.16·123-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1782272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1782272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1782272\)    =    \(2^{9} \cdot 59^{2}\)
Sign: $-1$
Analytic conductor: \(113.639\)
Root analytic conductor: \(3.26499\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1782272,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
59$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.3.ae_k
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.a_n
11$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.11.ac_x
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.13.a_z
17$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.17.c_bj
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.a_bq
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.37.a_cn
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.41.g_dn
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.43.c_dj
47$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.47.a_ag
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.a_dm
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.a_di
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.71.a_abb
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.79.a_gb
83$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.83.g_gt
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.89.ae_ha
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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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

−7.57348767771145521528098333038, −7.52358445330808110145092770460, −6.87449721853277266310712327746, −6.39634479409359963023127305588, −6.01322226126023577168666729028, −5.33821387574256957557153816185, −4.94470527400511720854522872417, −4.04834602375859777491821392157, −3.96895560779290292415517385473, −3.47490688399160712663053622399, −2.77953655047310056257929178464, −2.69851085698981014964242481632, −1.79192343081583970731074200923, −1.60368172404673005909935142882, 0, 1.60368172404673005909935142882, 1.79192343081583970731074200923, 2.69851085698981014964242481632, 2.77953655047310056257929178464, 3.47490688399160712663053622399, 3.96895560779290292415517385473, 4.04834602375859777491821392157, 4.94470527400511720854522872417, 5.33821387574256957557153816185, 6.01322226126023577168666729028, 6.39634479409359963023127305588, 6.87449721853277266310712327746, 7.52358445330808110145092770460, 7.57348767771145521528098333038

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