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Properties

Label 4-960e2-1.1-c1e2-0-0
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 − 16·19-s − 25-s − 12·29-s − 16·31-s + 12·41-s + 2·45-s − 2·49-s + 12·61-s − 32·71-s + 16·79-s + 81-s + 20·89-s + 32·95-s − 28·101-s − 20·109-s − 22·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s + 32·155-s + ⋯
L(s)  = 1  − 0.894·5-s − 1/3·9-s − 3.67·19-s − 1/5·25-s − 2.22·29-s − 2.87·31-s + 1.87·41-s + 0.298·45-s − 2/7·49-s + 1.53·61-s − 3.79·71-s + 1.80·79-s + 1/9·81-s + 2.11·89-s + 3.28·95-s − 2.78·101-s − 1.91·109-s − 2·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s + 2.57·155-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

\(L(1)\) \(\approx\) \(0.2455926919\)
\(L(\frac12)\) \(\approx\) \(0.2455926919\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.17.a_abi
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.19.q_dy
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.37.a_acg
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.47.a_ada
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.53.a_bm
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.67.a_k
71$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.71.bg_pi
73$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.73.a_afq
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.a_afa
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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

−10.64850060700796335976412945783, −9.607208219231976433725115211268, −9.394678421765754159672368337067, −8.905018687839236489232280382765, −8.642078670759440273002308176787, −8.090390279095231022283058379342, −7.82183629161162015228831571411, −7.18003304867701248986155467700, −7.08846359940981139173983656037, −6.30416802823338848306496242264, −5.95244055282415037170098171972, −5.58818040200031538797964585924, −4.95724440158262981913659660265, −4.21277196913622945465291462509, −3.97662250264514243236257060979, −3.75090853826914515243862014037, −2.84630717782602778042152556299, −2.06337812407088477085389490074, −1.84133745744896749087931043353, −0.21657302174663672672792344698, 0.21657302174663672672792344698, 1.84133745744896749087931043353, 2.06337812407088477085389490074, 2.84630717782602778042152556299, 3.75090853826914515243862014037, 3.97662250264514243236257060979, 4.21277196913622945465291462509, 4.95724440158262981913659660265, 5.58818040200031538797964585924, 5.95244055282415037170098171972, 6.30416802823338848306496242264, 7.08846359940981139173983656037, 7.18003304867701248986155467700, 7.82183629161162015228831571411, 8.090390279095231022283058379342, 8.642078670759440273002308176787, 8.905018687839236489232280382765, 9.394678421765754159672368337067, 9.607208219231976433725115211268, 10.64850060700796335976412945783

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