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Properties

Label 4-1344e2-1.1-c1e2-0-50
Degree $4$
Conductor $1806336$
Sign $-1$
Analytic cond. $115.173$
Root an. cond. $3.27595$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9-s − 2·11-s + 4·23-s − 2·25-s − 6·29-s − 2·37-s + 14·43-s − 7·49-s − 14·53-s + 18·67-s − 16·71-s + 4·79-s + 81-s − 2·99-s − 2·107-s − 10·109-s − 8·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 1/3·9-s − 0.603·11-s + 0.834·23-s − 2/5·25-s − 1.11·29-s − 0.328·37-s + 2.13·43-s − 49-s − 1.92·53-s + 2.19·67-s − 1.89·71-s + 0.450·79-s + 1/9·81-s − 0.201·99-s − 0.193·107-s − 0.957·109-s − 0.752·113-s + 6/11·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1806336\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(115.173\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1806336,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.c_ac
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.17.a_ba
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.ae_o
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.g_cg
31$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.31.a_ao
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.c_co
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.41.a_abi
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.43.ao_fe
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.o_fa
59$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \) 2.59.a_dq
61$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.61.a_c
67$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.67.as_ig
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.q_fm
73$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.73.a_abm
79$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.ae_ck
83$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.83.a_ade
89$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.89.a_be
97$C_2^2$ \( 1 + 142 T^{2} + p^{2} T^{4} \) 2.97.a_fm
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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

−7.64589356584638519621656218205, −7.09502129155599124603301507033, −6.93114218643772480262408292003, −6.24502149176916677075051351099, −5.87029448868965981253314090759, −5.46508321560339753948744935974, −4.92537914047315154619133471941, −4.58266607020609652820178252383, −3.99530873278142870022892327539, −3.50835424071167342504240715569, −2.98154913076128086542673020956, −2.39523984518311309620601124427, −1.80121942008171275671905320958, −1.06265784872373628984446803587, 0, 1.06265784872373628984446803587, 1.80121942008171275671905320958, 2.39523984518311309620601124427, 2.98154913076128086542673020956, 3.50835424071167342504240715569, 3.99530873278142870022892327539, 4.58266607020609652820178252383, 4.92537914047315154619133471941, 5.46508321560339753948744935974, 5.87029448868965981253314090759, 6.24502149176916677075051351099, 6.93114218643772480262408292003, 7.09502129155599124603301507033, 7.64589356584638519621656218205

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