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Properties

Label 4-1320e2-1.1-c1e2-0-20
Degree $4$
Conductor $1742400$
Sign $1$
Analytic cond. $111.096$
Root an. cond. $3.24657$
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  + 3·3-s − 2·4-s + 6·9-s − 6·12-s + 4·16-s − 10·19-s + 25-s + 9·27-s − 12·36-s + 4·43-s + 12·48-s + 5·49-s − 30·57-s − 8·64-s + 8·67-s + 20·73-s + 3·75-s + 20·76-s + 9·81-s + 12·97-s − 2·100-s − 18·108-s + 121-s + 127-s + 12·129-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.73·3-s − 4-s + 2·9-s − 1.73·12-s + 16-s − 2.29·19-s + 1/5·25-s + 1.73·27-s − 2·36-s + 0.609·43-s + 1.73·48-s + 5/7·49-s − 3.97·57-s − 64-s + 0.977·67-s + 2.34·73-s + 0.346·75-s + 2.29·76-s + 81-s + 1.21·97-s − 1/5·100-s − 1.73·108-s + 1/11·121-s + 0.0887·127-s + 1.05·129-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1742400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(111.096\)
Root analytic conductor: \(3.24657\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1742400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.933622057\)
\(L(\frac12)\) \(\approx\) \(2.933622057\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.7.a_af
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.a_z
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.19.k_cl
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.29.a_cf
31$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \) 2.31.a_acj
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.37.a_az
41$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.41.a_co
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.43.ae_dm
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.53.a_dt
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2^2$ \( 1 + 47 T^{2} + p^{2} T^{4} \) 2.61.a_bv
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.71.a_fd
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
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$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.89.a_abv
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.97.am_iw
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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

−8.042951703128934102033379744760, −7.59332520147809094571795914021, −7.07800426008146829374641699848, −6.68109144197669636382141169900, −6.12756610354418670105693511355, −5.70870461113209933596767769840, −4.86730114327834066731559580209, −4.73204334359590177178173372683, −4.01929190912749316951852479878, −3.85434630673308060398055054682, −3.37536387453007675861845473308, −2.61118707760511044268681237936, −2.26490424472686207864944297221, −1.63385937005350265595410279026, −0.65113737512512782605188263400, 0.65113737512512782605188263400, 1.63385937005350265595410279026, 2.26490424472686207864944297221, 2.61118707760511044268681237936, 3.37536387453007675861845473308, 3.85434630673308060398055054682, 4.01929190912749316951852479878, 4.73204334359590177178173372683, 4.86730114327834066731559580209, 5.70870461113209933596767769840, 6.12756610354418670105693511355, 6.68109144197669636382141169900, 7.07800426008146829374641699848, 7.59332520147809094571795914021, 8.042951703128934102033379744760

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