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Properties

Label 4-960e2-1.1-c1e2-0-29
Degree $4$
Conductor $921600$
Sign $1$
Analytic cond. $58.7620$
Root an. cond. $2.76868$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 9-s + 4·13-s + 4·17-s + 3·25-s + 4·29-s + 20·37-s + 20·41-s − 2·45-s − 14·49-s + 20·53-s + 4·61-s − 8·65-s + 20·73-s + 81-s − 8·85-s − 12·89-s + 4·97-s − 12·101-s − 28·109-s + 4·113-s + 4·117-s − 6·121-s − 4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.894·5-s + 1/3·9-s + 1.10·13-s + 0.970·17-s + 3/5·25-s + 0.742·29-s + 3.28·37-s + 3.12·41-s − 0.298·45-s − 2·49-s + 2.74·53-s + 0.512·61-s − 0.992·65-s + 2.34·73-s + 1/9·81-s − 0.867·85-s − 1.27·89-s + 0.406·97-s − 1.19·101-s − 2.68·109-s + 0.376·113-s + 0.369·117-s − 0.545·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(921600\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(58.7620\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 921600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.235701712\)
\(L(\frac12)\) \(\approx\) \(2.235701712\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$ \( ( 1 + T )^{2} \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.37.au_gs
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.41.au_ha
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.53.au_hy
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.a_ak
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−8.074746559817531846840486780032, −7.67749235610857150327119620769, −7.60125180356652570547951985370, −6.67875901406960725991694880905, −6.56365355473852823699216167747, −5.85130665991850657402735978605, −5.63406482120703248001185629214, −4.87700802399738744801917392626, −4.38871739502786245271092399731, −3.82349173075266898382654912582, −3.76558570458913747879946495025, −2.63252540175087701435373844808, −2.61544872003966305633910980827, −1.18815390030287753587945796282, −0.870154925974925884967934945032, 0.870154925974925884967934945032, 1.18815390030287753587945796282, 2.61544872003966305633910980827, 2.63252540175087701435373844808, 3.76558570458913747879946495025, 3.82349173075266898382654912582, 4.38871739502786245271092399731, 4.87700802399738744801917392626, 5.63406482120703248001185629214, 5.85130665991850657402735978605, 6.56365355473852823699216167747, 6.67875901406960725991694880905, 7.60125180356652570547951985370, 7.67749235610857150327119620769, 8.074746559817531846840486780032

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