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Properties

Label 4-20224-1.1-c1e2-0-1
Degree $4$
Conductor $20224$
Sign $1$
Analytic cond. $1.28949$
Root an. cond. $1.06562$
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  + 2·7-s + 4·9-s − 2·17-s − 14·23-s + 6·25-s + 10·31-s − 10·49-s + 8·63-s − 10·71-s + 20·73-s + 5·79-s + 7·81-s − 4·89-s − 8·97-s − 22·103-s + 4·113-s − 4·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s − 28·161-s + ⋯
L(s)  = 1  + 0.755·7-s + 4/3·9-s − 0.485·17-s − 2.91·23-s + 6/5·25-s + 1.79·31-s − 1.42·49-s + 1.00·63-s − 1.18·71-s + 2.34·73-s + 0.562·79-s + 7/9·81-s − 0.423·89-s − 0.812·97-s − 2.16·103-s + 0.376·113-s − 0.366·119-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.646·153-s + 0.0798·157-s − 2.20·161-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(20224\)    =    \(2^{8} \cdot 79\)
Sign: $1$
Analytic conductor: \(1.28949\)
Root analytic conductor: \(1.06562\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 20224,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.315641745\)
\(L(\frac12)\) \(\approx\) \(1.315641745\)
\(L(\frac{3}{2})\) 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 \( 1 \)
79$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 4 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.7.ac_o
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.11.a_as
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.13.a_o
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.c_k
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.o_dq
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.29.a_k
31$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.31.ak_di
37$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.37.a_abe
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.43.a_u
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.53.a_cg
59$C_2^2$ \( 1 + 116 T^{2} + p^{2} T^{4} \) 2.59.a_em
61$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.61.a_de
67$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \) 2.67.a_dm
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.71.k_fm
73$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.73.au_ic
83$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.83.a_ck
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.89.e_gw
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.i_o
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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

−10.82279930435305211692603160167, −10.23216883671663035065568895577, −9.903538042895832913925614850729, −9.396305240779335718859512404368, −8.532645017918498695329592976366, −8.041960823998210450974975635023, −7.76446220370725607266098283428, −6.79905070656406231807629998775, −6.51428122723939389207794343670, −5.70764072829959512030828183213, −4.76317549124031896576746886641, −4.43234981858358336968409206469, −3.69744280745966055623581423592, −2.43960072727405861259475321097, −1.49963739880155802822922737958, 1.49963739880155802822922737958, 2.43960072727405861259475321097, 3.69744280745966055623581423592, 4.43234981858358336968409206469, 4.76317549124031896576746886641, 5.70764072829959512030828183213, 6.51428122723939389207794343670, 6.79905070656406231807629998775, 7.76446220370725607266098283428, 8.041960823998210450974975635023, 8.532645017918498695329592976366, 9.396305240779335718859512404368, 9.903538042895832913925614850729, 10.23216883671663035065568895577, 10.82279930435305211692603160167

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