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Properties

Label 4-1782272-1.1-c1e2-0-1
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·9-s − 12·17-s + 6·19-s − 9·25-s + 14·27-s − 10·41-s − 13·49-s + 24·51-s − 12·57-s + 2·59-s + 8·67-s − 12·73-s + 18·75-s − 4·81-s − 28·83-s + 24·89-s + 12·97-s + 6·107-s − 16·113-s − 22·121-s + 20·123-s + 127-s + 131-s + 137-s + 139-s + 26·147-s + ⋯
L(s)  = 1  − 1.15·3-s − 9-s − 2.91·17-s + 1.37·19-s − 9/5·25-s + 2.69·27-s − 1.56·41-s − 1.85·49-s + 3.36·51-s − 1.58·57-s + 0.260·59-s + 0.977·67-s − 1.40·73-s + 2.07·75-s − 4/9·81-s − 3.07·83-s + 2.54·89-s + 1.21·97-s + 0.580·107-s − 1.50·113-s − 2·121-s + 1.80·123-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.14·147-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: \(2\)
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$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
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$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.29.a_bx
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$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.41.k_ed
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.a_dm
53$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.53.a_dt
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.a_aw
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.79.a_acp
83$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.83.bc_ny
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.89.ay_mk
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.97.am_iw
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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

−7.23503296602818369515270586627, −6.84574257958418937660779698825, −6.43861163420690481107923289495, −6.05909468696688153232644342575, −5.78361551267959519878472884800, −5.15803147848862332612722517294, −4.88115795806245545189133936314, −4.49694535208075381432652481944, −3.73646381098494648738609860838, −3.31473196076294193224612457762, −2.57873205237227455984509773778, −2.17711837566398295927851871802, −1.32843621990161482531558900839, 0, 0, 1.32843621990161482531558900839, 2.17711837566398295927851871802, 2.57873205237227455984509773778, 3.31473196076294193224612457762, 3.73646381098494648738609860838, 4.49694535208075381432652481944, 4.88115795806245545189133936314, 5.15803147848862332612722517294, 5.78361551267959519878472884800, 6.05909468696688153232644342575, 6.43861163420690481107923289495, 6.84574257958418937660779698825, 7.23503296602818369515270586627

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