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Properties

Label 4-1782272-1.1-c1e2-0-0
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  − 2·3-s − 3·9-s + 8·11-s + 4·17-s + 6·19-s − 9·25-s + 14·27-s − 16·33-s + 6·41-s + 16·43-s − 13·49-s − 8·51-s − 12·57-s + 2·59-s − 16·67-s − 12·73-s + 18·75-s − 4·81-s + 12·83-s − 32·89-s − 20·97-s − 24·99-s + 6·107-s + 16·113-s + 26·121-s − 12·123-s + 127-s + ⋯
L(s)  = 1  − 1.15·3-s − 9-s + 2.41·11-s + 0.970·17-s + 1.37·19-s − 9/5·25-s + 2.69·27-s − 2.78·33-s + 0.937·41-s + 2.43·43-s − 1.85·49-s − 1.12·51-s − 1.58·57-s + 0.260·59-s − 1.95·67-s − 1.40·73-s + 2.07·75-s − 4/9·81-s + 1.31·83-s − 3.39·89-s − 2.03·97-s − 2.41·99-s + 0.580·107-s + 1.50·113-s + 2.36·121-s − 1.08·123-s + 0.0887·127-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$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.3.c_h
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.a_j
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.a_n
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.11.ai_bm
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.19.ag_bv
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.29.a_bh
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.41.ag_dn
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.a_dm
53$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.53.a_ap
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.67.q_hq
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
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$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.89.bg_qs
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

−7.40961699026055564212526317651, −7.29096976680608652163781804176, −6.40387843928083411347729073969, −6.30393778815316113589319798048, −5.94264930407483059532376201400, −5.39735883253081236426184867578, −5.36336736026187235479038459757, −4.28681391729088540393374354665, −4.26591491377627131499860476385, −3.49734312353215216887343182177, −3.08925969827549501011165766342, −2.46296394936642547160226286791, −1.36824680440498460384564598194, −1.11397760025351109152581002959, 0, 1.11397760025351109152581002959, 1.36824680440498460384564598194, 2.46296394936642547160226286791, 3.08925969827549501011165766342, 3.49734312353215216887343182177, 4.26591491377627131499860476385, 4.28681391729088540393374354665, 5.36336736026187235479038459757, 5.39735883253081236426184867578, 5.94264930407483059532376201400, 6.30393778815316113589319798048, 6.40387843928083411347729073969, 7.29096976680608652163781804176, 7.40961699026055564212526317651

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