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Properties

Label 4-1184e2-1.1-c1e2-0-8
Degree $4$
Conductor $1401856$
Sign $-1$
Analytic cond. $89.3835$
Root an. cond. $3.07478$
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  + 4·5-s − 6·9-s + 4·13-s + 4·17-s + 2·25-s − 12·29-s − 2·37-s − 4·41-s − 24·45-s + 2·49-s − 12·53-s − 12·61-s + 16·65-s + 28·73-s + 27·81-s + 16·85-s − 12·89-s + 20·97-s − 12·101-s + 4·109-s + 20·113-s − 24·117-s − 22·121-s − 28·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.78·5-s − 2·9-s + 1.10·13-s + 0.970·17-s + 2/5·25-s − 2.22·29-s − 0.328·37-s − 0.624·41-s − 3.57·45-s + 2/7·49-s − 1.64·53-s − 1.53·61-s + 1.98·65-s + 3.27·73-s + 3·81-s + 1.73·85-s − 1.27·89-s + 2.03·97-s − 1.19·101-s + 0.383·109-s + 1.88·113-s − 2.21·117-s − 2·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1401856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1401856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1401856\)    =    \(2^{10} \cdot 37^{2}\)
Sign: $-1$
Analytic conductor: \(89.3835\)
Root analytic conductor: \(3.07478\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1401856,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
37$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.5.ae_o
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.a_bi
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.a_cg
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.43.a_de
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.a_da
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.61.m_gc
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.a_cs
71$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.71.a_ew
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.73.abc_ne
79$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.79.a_fy
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_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

−7.75719220170824942128020627833, −7.44643679398612564054429895074, −6.53662216607494549214962948413, −6.22415831207096217368749210189, −6.04755779498159347807544806648, −5.46349139552603568969562413832, −5.41123543087893400462645312371, −4.88253984327955854319240680173, −3.85223878389055117023714736235, −3.50211556528427241737853655370, −3.06051561029494874253606536430, −2.30384470940550542264449691743, −1.93554368244305693557513075885, −1.26629318215268350130600876569, 0, 1.26629318215268350130600876569, 1.93554368244305693557513075885, 2.30384470940550542264449691743, 3.06051561029494874253606536430, 3.50211556528427241737853655370, 3.85223878389055117023714736235, 4.88253984327955854319240680173, 5.41123543087893400462645312371, 5.46349139552603568969562413832, 6.04755779498159347807544806648, 6.22415831207096217368749210189, 6.53662216607494549214962948413, 7.44643679398612564054429895074, 7.75719220170824942128020627833

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