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Properties

Label 4-1184e2-1.1-c1e2-0-7
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·5-s + 3·9-s + 12·13-s − 8·17-s + 38·25-s − 8·29-s − 2·37-s − 10·41-s + 24·45-s − 5·49-s + 10·53-s − 20·61-s + 96·65-s + 6·73-s − 64·85-s − 4·89-s − 12·97-s − 10·101-s − 12·109-s + 8·113-s + 36·117-s − 13·121-s + 136·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 3.57·5-s + 9-s + 3.32·13-s − 1.94·17-s + 38/5·25-s − 1.48·29-s − 0.328·37-s − 1.56·41-s + 3.57·45-s − 5/7·49-s + 1.37·53-s − 2.56·61-s + 11.9·65-s + 0.702·73-s − 6.94·85-s − 0.423·89-s − 1.21·97-s − 0.995·101-s − 1.14·109-s + 0.752·113-s + 3.32·117-s − 1.18·121-s + 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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: \(0\)
Selberg data: \((4,\ 1401856,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.974154499\)
\(L(\frac12)\) \(\approx\) \(5.974154499\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.3.a_ad
5$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.5.ai_ba
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.a_n
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.17.i_by
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.a_c
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.29.i_cw
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.41.k_ed
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.43.a_by
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.a_n
53$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.53.ak_fb
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.a_ak
71$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.71.a_cj
73$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.73.ag_fz
79$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.79.a_o
83$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.83.a_gb
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.89.e_ha
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.97.m_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

−8.171116173413720852515064409422, −7.27498861153339197087765218331, −6.70215534378769498619674725507, −6.43518047637612164386054425282, −6.37130663081964170851026344460, −5.77142731733412237415577053303, −5.50386762915442186383450566417, −5.07368852733938728543400465493, −4.35244050864138721227666504237, −3.89119998683865980446546080420, −3.20217122267469588133159845598, −2.54151943185940669283450208208, −1.80714683000640168781195911291, −1.64599982063715737449109612262, −1.25838069591403157091036114514, 1.25838069591403157091036114514, 1.64599982063715737449109612262, 1.80714683000640168781195911291, 2.54151943185940669283450208208, 3.20217122267469588133159845598, 3.89119998683865980446546080420, 4.35244050864138721227666504237, 5.07368852733938728543400465493, 5.50386762915442186383450566417, 5.77142731733412237415577053303, 6.37130663081964170851026344460, 6.43518047637612164386054425282, 6.70215534378769498619674725507, 7.27498861153339197087765218331, 8.171116173413720852515064409422

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