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Properties

Label 4-2112e2-1.1-c1e2-0-28
Degree $4$
Conductor $4460544$
Sign $1$
Analytic cond. $284.408$
Root an. cond. $4.10662$
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  − 2·3-s + 9-s − 6·11-s + 8·25-s + 4·27-s + 12·29-s + 8·31-s + 12·33-s − 4·37-s + 12·41-s − 4·49-s − 8·67-s − 16·75-s − 11·81-s + 24·83-s − 24·87-s − 16·93-s + 16·97-s − 6·99-s + 12·101-s + 8·103-s + 36·107-s + 8·111-s + 25·121-s − 24·123-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 1.80·11-s + 8/5·25-s + 0.769·27-s + 2.22·29-s + 1.43·31-s + 2.08·33-s − 0.657·37-s + 1.87·41-s − 4/7·49-s − 0.977·67-s − 1.84·75-s − 1.22·81-s + 2.63·83-s − 2.57·87-s − 1.65·93-s + 1.62·97-s − 0.603·99-s + 1.19·101-s + 0.788·103-s + 3.48·107-s + 0.759·111-s + 2.27·121-s − 2.16·123-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4460544\)    =    \(2^{12} \cdot 3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(284.408\)
Root analytic conductor: \(4.10662\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4460544,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.515299114\)
\(L(\frac12)\) \(\approx\) \(1.515299114\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
11$C_2$ \( 1 + 6 T + p T^{2} \)
good5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.5.a_ai
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.13.a_ai
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.19.a_abm
23$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.23.a_abs
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.47.a_e
53$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \) 2.53.a_ace
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.59.a_k
61$C_2^2$ \( 1 - 104 T^{2} + p^{2} T^{4} \) 2.61.a_aea
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2^2$ \( 1 - 92 T^{2} + p^{2} T^{4} \) 2.71.a_ado
73$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.73.a_afq
79$C_2^2$ \( 1 - 140 T^{2} + p^{2} T^{4} \) 2.79.a_afk
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.83.ay_ly
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.a_afq
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.97.aq_jy
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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

−9.071029150076800361562111613650, −8.977717955324590207082064631822, −8.542151764652393342309668719335, −7.935958184105677884721674853721, −7.87820071412967745139675770346, −7.36508293739111495011204226340, −6.80172812138180899082851824641, −6.46819314286450874364517234410, −6.17978252182744207197756517870, −5.76379870002548448199234651693, −5.19805930650152101498565515551, −4.92843913860602747226894358864, −4.63061461280788769521071676917, −4.31028720755800947733238182263, −3.21887960503175479196204933222, −3.06522245225528067057785414095, −2.56051947352174683859975453295, −1.97168572172222986493942571417, −0.76940989515886303716597835661, −0.72525559340748650857202128611, 0.72525559340748650857202128611, 0.76940989515886303716597835661, 1.97168572172222986493942571417, 2.56051947352174683859975453295, 3.06522245225528067057785414095, 3.21887960503175479196204933222, 4.31028720755800947733238182263, 4.63061461280788769521071676917, 4.92843913860602747226894358864, 5.19805930650152101498565515551, 5.76379870002548448199234651693, 6.17978252182744207197756517870, 6.46819314286450874364517234410, 6.80172812138180899082851824641, 7.36508293739111495011204226340, 7.87820071412967745139675770346, 7.935958184105677884721674853721, 8.542151764652393342309668719335, 8.977717955324590207082064631822, 9.071029150076800361562111613650

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