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Properties

Label 4-4080e2-1.1-c1e2-0-8
Degree $4$
Conductor $16646400$
Sign $1$
Analytic cond. $1061.38$
Root an. cond. $5.70779$
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  − 9-s − 4·13-s + 2·17-s − 6·19-s − 25-s + 8·43-s − 14·47-s + 5·49-s − 2·53-s + 28·67-s + 81-s + 32·83-s + 16·89-s + 24·101-s − 4·103-s + 4·117-s + 13·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2·153-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1/3·9-s − 1.10·13-s + 0.485·17-s − 1.37·19-s − 1/5·25-s + 1.21·43-s − 2.04·47-s + 5/7·49-s − 0.274·53-s + 3.42·67-s + 1/9·81-s + 3.51·83-s + 1.69·89-s + 2.38·101-s − 0.394·103-s + 0.369·117-s + 1.18·121-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 − 0.161·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16646400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16646400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16646400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1061.38\)
Root analytic conductor: \(5.70779\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16646400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.717987056\)
\(L(\frac12)\) \(\approx\) \(1.717987056\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + T^{2} \)
17$C_2$ \( 1 - 2 T + p T^{2} \)
good7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.7.a_af
11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.11.a_an
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
19$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.19.g_bv
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.23.a_s
29$C_2^2$ \( 1 - 57 T^{2} + p^{2} T^{4} \) 2.29.a_acf
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.31.a_c
37$C_2^2$ \( 1 - 49 T^{2} + p^{2} T^{4} \) 2.37.a_abx
41$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \) 2.41.a_acv
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.47.o_fn
53$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.53.c_ed
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.61.a_aes
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.67.abc_ms
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.71.a_c
73$C_2^2$ \( 1 - 145 T^{2} + p^{2} T^{4} \) 2.73.a_afp
79$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \) 2.79.a_afy
83$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.83.abg_qg
89$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.89.aq_ji
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.97.a_adq
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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

−8.675384334165728676510446664704, −8.268589431134327844546454461965, −7.70633803944749711034125703785, −7.68221876335684161538443157909, −7.28481250713467774624553168335, −6.52423080951018741557147171601, −6.49475463201529702524408320851, −6.24873716954586493203556356462, −5.52113463609601605149037438914, −5.27068454661324805654277617250, −4.89211220257082829283662556724, −4.53606063478198043933885251610, −4.05524264944778685382928604950, −3.47764141289984447967550198570, −3.36568070754930469029056588118, −2.55620436763234161304050576658, −2.16000282834970843463843863785, −1.99975018061382739993829951365, −0.990401804663595607157165163897, −0.42653798816578584800145683481, 0.42653798816578584800145683481, 0.990401804663595607157165163897, 1.99975018061382739993829951365, 2.16000282834970843463843863785, 2.55620436763234161304050576658, 3.36568070754930469029056588118, 3.47764141289984447967550198570, 4.05524264944778685382928604950, 4.53606063478198043933885251610, 4.89211220257082829283662556724, 5.27068454661324805654277617250, 5.52113463609601605149037438914, 6.24873716954586493203556356462, 6.49475463201529702524408320851, 6.52423080951018741557147171601, 7.28481250713467774624553168335, 7.68221876335684161538443157909, 7.70633803944749711034125703785, 8.268589431134327844546454461965, 8.675384334165728676510446664704

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