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Properties

Label 4-5888-1.1-c1e2-0-1
Degree $4$
Conductor $5888$
Sign $1$
Analytic cond. $0.375423$
Root an. cond. $0.782763$
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  + 2·9-s − 3·23-s + 2·25-s − 12·31-s + 4·41-s − 14·49-s + 8·71-s − 12·73-s + 4·79-s − 5·81-s + 8·89-s − 8·97-s + 4·103-s + 28·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 2/3·9-s − 0.625·23-s + 2/5·25-s − 2.15·31-s + 0.624·41-s − 2·49-s + 0.949·71-s − 1.40·73-s + 0.450·79-s − 5/9·81-s + 0.847·89-s − 0.812·97-s + 0.394·103-s + 2.63·113-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 − 0.461·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5888 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5888 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5888\)    =    \(2^{8} \cdot 23\)
Sign: $1$
Analytic conductor: \(0.375423\)
Root analytic conductor: \(0.782763\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5888,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8904167946\)
\(L(\frac12)\) \(\approx\) \(0.8904167946\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
29$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.29.a_aba
31$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.m_dq
37$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.37.a_ac
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.ae_w
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.43.a_cg
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.53.a_be
59$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.59.a_acw
61$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.61.a_ck
67$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.67.a_ba
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.ai_fm
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.m_gk
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.79.ae_gc
83$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.83.a_ba
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.ai_dq
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.97.i_hy
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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

−12.21243640515196138927564339907, −11.26263024840018002317689050890, −11.12675226206825921642794322576, −10.24719885819682767407229383392, −9.812454172558123444989827272005, −9.163680712720856373875142972702, −8.572637785949946620702674519742, −7.74865725037914944185292859112, −7.28662807515928418026871184804, −6.53488364177421238947333125610, −5.79222931109338987216594943657, −4.97771703872054360824409504004, −4.15412424799406019272273144942, −3.27618472930653916328746805117, −1.87091752077799971231492786390, 1.87091752077799971231492786390, 3.27618472930653916328746805117, 4.15412424799406019272273144942, 4.97771703872054360824409504004, 5.79222931109338987216594943657, 6.53488364177421238947333125610, 7.28662807515928418026871184804, 7.74865725037914944185292859112, 8.572637785949946620702674519742, 9.163680712720856373875142972702, 9.812454172558123444989827272005, 10.24719885819682767407229383392, 11.12675226206825921642794322576, 11.26263024840018002317689050890, 12.21243640515196138927564339907

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