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Properties

Label 4-168e2-1.1-c1e2-0-7
Degree $4$
Conductor $28224$
Sign $1$
Analytic cond. $1.79958$
Root an. cond. $1.15822$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·4-s + 9-s − 4·12-s + 4·16-s + 12·19-s + 6·25-s − 4·27-s − 2·36-s − 4·43-s + 8·48-s − 49-s + 24·57-s − 8·64-s − 4·67-s − 12·73-s + 12·75-s − 24·76-s − 11·81-s − 4·97-s − 12·100-s + 8·108-s − 18·121-s + 127-s − 8·129-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.15·3-s − 4-s + 1/3·9-s − 1.15·12-s + 16-s + 2.75·19-s + 6/5·25-s − 0.769·27-s − 1/3·36-s − 0.609·43-s + 1.15·48-s − 1/7·49-s + 3.17·57-s − 64-s − 0.488·67-s − 1.40·73-s + 1.38·75-s − 2.75·76-s − 1.22·81-s − 0.406·97-s − 6/5·100-s + 0.769·108-s − 1.63·121-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(28224\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1.79958\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 28224,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.472718581\)
\(L(\frac12)\) \(\approx\) \(1.472718581\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
3$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.19.am_cw
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.a_ag
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.31.a_c
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.43.e_dm
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.59.a_s
61$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.61.a_aec
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.67.e_fi
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.79.a_adq
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
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.e_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

−10.28620399341622470550638453054, −9.887136630325027490666649135811, −9.393507695232222751319006863113, −9.002136739186055216164212439106, −8.567329421530738796335254021138, −7.938646266925342778705410318895, −7.53208109086882074354712245319, −6.98762414182075845869423032524, −5.99375492422382120737687149168, −5.31306434220500476035379994936, −4.87550560097527312407599615644, −3.96019251490424761722410398205, −3.27001075285133692901361970260, −2.82526965718183128765437634941, −1.30990847497478007676972301682, 1.30990847497478007676972301682, 2.82526965718183128765437634941, 3.27001075285133692901361970260, 3.96019251490424761722410398205, 4.87550560097527312407599615644, 5.31306434220500476035379994936, 5.99375492422382120737687149168, 6.98762414182075845869423032524, 7.53208109086882074354712245319, 7.938646266925342778705410318895, 8.567329421530738796335254021138, 9.002136739186055216164212439106, 9.393507695232222751319006863113, 9.887136630325027490666649135811, 10.28620399341622470550638453054

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