Properties

Label 4-1776e2-1.1-c1e2-0-20
Degree $4$
Conductor $3154176$
Sign $1$
Analytic cond. $201.112$
Root an. cond. $3.76582$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 6·5-s + 7-s − 12·11-s + 3·13-s − 6·15-s + 12·17-s − 6·19-s + 21-s + 19·25-s − 27-s − 12·33-s − 6·35-s − 10·37-s + 3·39-s − 12·41-s − 24·47-s + 7·49-s + 12·51-s − 6·53-s + 72·55-s − 6·57-s + 6·59-s + 12·61-s − 18·65-s − 5·67-s + 14·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 2.68·5-s + 0.377·7-s − 3.61·11-s + 0.832·13-s − 1.54·15-s + 2.91·17-s − 1.37·19-s + 0.218·21-s + 19/5·25-s − 0.192·27-s − 2.08·33-s − 1.01·35-s − 1.64·37-s + 0.480·39-s − 1.87·41-s − 3.50·47-s + 49-s + 1.68·51-s − 0.824·53-s + 9.70·55-s − 0.794·57-s + 0.781·59-s + 1.53·61-s − 2.23·65-s − 0.610·67-s + 1.63·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3154176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3154176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3154176\)    =    \(2^{8} \cdot 3^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(201.112\)
Root analytic conductor: \(3.76582\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3154176,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3$C_2$ \( 1 - T + T^{2} \)
37$C_2$ \( 1 + 10 T + p T^{2} \)
good5$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.5.g_r
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.ab_ag
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.11.m_cg
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.ad_q
17$C_2^2$ \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.17.am_cn
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.g_bf
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.23.a_abi
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.a_ach
41$C_2^2$ \( 1 + 12 T + 103 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_dz
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.a_cj
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.47.y_je
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_ar
59$C_2^2$ \( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_ct
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.am_ef
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.f_abq
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.71.a_act
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.73.ao_hn
79$C_2^2$ \( 1 + 3 T + 82 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.79.d_de
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_cj
89$C_2^2$ \( 1 - 18 T + 197 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.89.as_hp
97$C_2^2$ \( 1 - 191 T^{2} + p^{2} T^{4} \) 2.97.a_ahj
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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

−8.698040574203444117350388076744, −8.284240739467849834831200801926, −8.271886193440367420909765847762, −7.995569572111011164870962261870, −7.65576053407212149437387746036, −7.56857163183667331135319514843, −6.86180475129778439787213227881, −6.47860027730196002756742771664, −5.59800017376546958120928233100, −5.25353694314861158507706981204, −4.98473062494809732778200194214, −4.68470205554538382148836130067, −3.65907766483328818489774049568, −3.59629402985586351891222112675, −3.36291910475398319502504128260, −2.74362771514290852962528570037, −2.17272926869154637850649909301, −1.20503928985695502022872761724, 0, 0, 1.20503928985695502022872761724, 2.17272926869154637850649909301, 2.74362771514290852962528570037, 3.36291910475398319502504128260, 3.59629402985586351891222112675, 3.65907766483328818489774049568, 4.68470205554538382148836130067, 4.98473062494809732778200194214, 5.25353694314861158507706981204, 5.59800017376546958120928233100, 6.47860027730196002756742771664, 6.86180475129778439787213227881, 7.56857163183667331135319514843, 7.65576053407212149437387746036, 7.995569572111011164870962261870, 8.271886193440367420909765847762, 8.284240739467849834831200801926, 8.698040574203444117350388076744

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