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Properties

Label 4-105e2-1.1-c1e2-0-2
Degree $4$
Conductor $11025$
Sign $1$
Analytic cond. $0.702963$
Root an. cond. $0.915657$
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  + 3-s − 4·4-s + 2·7-s − 2·9-s − 4·12-s + 10·13-s + 12·16-s + 4·19-s + 2·21-s + 25-s − 5·27-s − 8·28-s − 8·31-s + 8·36-s + 4·37-s + 10·39-s − 20·43-s + 12·48-s + 3·49-s − 40·52-s + 4·57-s + 16·61-s − 4·63-s − 32·64-s − 8·67-s + 4·73-s + 75-s + ⋯
L(s)  = 1  + 0.577·3-s − 2·4-s + 0.755·7-s − 2/3·9-s − 1.15·12-s + 2.77·13-s + 3·16-s + 0.917·19-s + 0.436·21-s + 1/5·25-s − 0.962·27-s − 1.51·28-s − 1.43·31-s + 4/3·36-s + 0.657·37-s + 1.60·39-s − 3.04·43-s + 1.73·48-s + 3/7·49-s − 5.54·52-s + 0.529·57-s + 2.04·61-s − 0.503·63-s − 4·64-s − 0.977·67-s + 0.468·73-s + 0.115·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11025\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.702963\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8948642832\)
\(L(\frac12)\) \(\approx\) \(0.8948642832\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3$C_2$ \( 1 - T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.2.a_e
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.a_n
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.13.ak_bz
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.a_z
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.29.a_bx
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.a_ack
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.43.u_he
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.a_n
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.a_abm
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.61.aq_he
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.79.c_gd
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.a_bi
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.97.c_hn
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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

−11.50769791186352205274303435148, −10.81642134200306905635056067311, −10.26175613408033680908306725927, −9.502060864672765915422726581101, −9.001872818363131182920485953111, −8.679498853252053444462430241617, −8.087023964728641191414598204496, −7.980947296320422939728903060714, −6.69771158952637091109528432278, −5.63914839037632160871019437372, −5.47907564040983295563221596641, −4.54321060660836499326534703030, −3.61689003045514298236484607491, −3.45067056060882590557565434961, −1.35738749247200780270220785705, 1.35738749247200780270220785705, 3.45067056060882590557565434961, 3.61689003045514298236484607491, 4.54321060660836499326534703030, 5.47907564040983295563221596641, 5.63914839037632160871019437372, 6.69771158952637091109528432278, 7.980947296320422939728903060714, 8.087023964728641191414598204496, 8.679498853252053444462430241617, 9.001872818363131182920485953111, 9.502060864672765915422726581101, 10.26175613408033680908306725927, 10.81642134200306905635056067311, 11.50769791186352205274303435148

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