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Properties

Label 4-210e2-1.1-c1e2-0-16
Degree $4$
Conductor $44100$
Sign $-1$
Analytic cond. $2.81185$
Root an. cond. $1.29493$
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-s − 2·7-s − 3·9-s − 12·13-s + 16-s + 25-s − 2·28-s + 16·31-s − 3·36-s − 20·37-s + 8·43-s + 3·49-s − 12·52-s − 28·61-s + 6·63-s + 64-s − 24·67-s + 4·73-s − 16·79-s + 9·81-s + 24·91-s + 4·97-s + 100-s + 32·103-s + 12·109-s − 2·112-s + 36·117-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.755·7-s − 9-s − 3.32·13-s + 1/4·16-s + 1/5·25-s − 0.377·28-s + 2.87·31-s − 1/2·36-s − 3.28·37-s + 1.21·43-s + 3/7·49-s − 1.66·52-s − 3.58·61-s + 0.755·63-s + 1/8·64-s − 2.93·67-s + 0.468·73-s − 1.80·79-s + 81-s + 2.51·91-s + 0.406·97-s + 1/10·100-s + 3.15·103-s + 1.14·109-s − 0.188·112-s + 3.32·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(44100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(2.81185\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 44100,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( ( 1 + T )^{2} \)
good11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.13.m_ck
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.a_da
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.a_dy
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.a_cc
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.61.bc_mg
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.a_aek
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.83.a_dy
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
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

−10.06791593868518168838717297188, −9.463210187800135113179931417957, −8.977646092372605556420891081568, −8.454018593614137368858882513975, −7.52893955472642377706219869463, −7.45420248729985818641828193491, −6.80466003204333393578640397117, −6.17616144092315208416665412995, −5.64303042849579726385302435090, −4.73779460478605507715027170236, −4.63923341949934829303628265619, −3.00898354745033704314501957111, −3.00381372509091040158022668795, −2.08529359572419276696314052557, 0, 2.08529359572419276696314052557, 3.00381372509091040158022668795, 3.00898354745033704314501957111, 4.63923341949934829303628265619, 4.73779460478605507715027170236, 5.64303042849579726385302435090, 6.17616144092315208416665412995, 6.80466003204333393578640397117, 7.45420248729985818641828193491, 7.52893955472642377706219869463, 8.454018593614137368858882513975, 8.977646092372605556420891081568, 9.463210187800135113179931417957, 10.06791593868518168838717297188

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