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Properties

Label 4-756e2-1.1-c1e2-0-2
Degree $4$
Conductor $571536$
Sign $1$
Analytic cond. $36.4416$
Root an. cond. $2.45696$
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  − 4·7-s − 25-s − 2·37-s − 20·43-s + 9·49-s + 4·67-s + 28·79-s + 22·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 173-s + 4·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 1.51·7-s − 1/5·25-s − 0.328·37-s − 3.04·43-s + 9/7·49-s + 0.488·67-s + 3.15·79-s + 2.10·109-s + 5/11·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.07·169-s + 0.0760·173-s + 0.302·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(571536\)    =    \(2^{4} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(36.4416\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 571536,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.050170418\)
\(L(\frac12)\) \(\approx\) \(1.050170418\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.11.a_af
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.13.a_ao
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.19.a_abj
23$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.23.a_t
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.29.a_aby
31$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.31.a_abj
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.37.c_cx
41$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.41.a_cv
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.43.u_he
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.61.a_cs
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.67.ae_fi
71$C_2^2$ \( 1 + 115 T^{2} + p^{2} T^{4} \) 2.71.a_el
73$C_2^2$ \( 1 - 134 T^{2} + p^{2} T^{4} \) 2.73.a_afe
79$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.79.abc_nq
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.89.a_dt
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \) 2.97.a_afq
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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

−8.508223105035155406740876662947, −7.972054802391278854810172128354, −7.55619236906060410148899061028, −6.84438419074630738001060084198, −6.72059817438280470988599638852, −6.23608013684021156446636446687, −5.78665757482908124943983848048, −5.10476320968109623770249342776, −4.81387435522668107618366033053, −3.96931171960197304206812975638, −3.45858780655776907678316161835, −3.20064445851251945725995627989, −2.39832020689399627116489264627, −1.70640546426418867540413705485, −0.51498536895014379252286254345, 0.51498536895014379252286254345, 1.70640546426418867540413705485, 2.39832020689399627116489264627, 3.20064445851251945725995627989, 3.45858780655776907678316161835, 3.96931171960197304206812975638, 4.81387435522668107618366033053, 5.10476320968109623770249342776, 5.78665757482908124943983848048, 6.23608013684021156446636446687, 6.72059817438280470988599638852, 6.84438419074630738001060084198, 7.55619236906060410148899061028, 7.972054802391278854810172128354, 8.508223105035155406740876662947

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