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Properties

Label 4-88e2-1.1-c1e2-0-2
Degree $4$
Conductor $7744$
Sign $1$
Analytic cond. $0.493764$
Root an. cond. $0.838262$
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  + 4·3-s − 2·4-s + 6·9-s − 6·11-s − 8·12-s + 4·16-s + 10·25-s − 4·27-s − 24·33-s − 12·36-s + 12·44-s + 16·48-s − 14·49-s − 12·59-s − 8·64-s + 28·67-s + 40·75-s − 37·81-s + 36·89-s + 20·97-s − 36·99-s − 20·100-s + 8·108-s − 36·113-s + 25·121-s + 127-s + 131-s + ⋯
L(s)  = 1  + 2.30·3-s − 4-s + 2·9-s − 1.80·11-s − 2.30·12-s + 16-s + 2·25-s − 0.769·27-s − 4.17·33-s − 2·36-s + 1.80·44-s + 2.30·48-s − 2·49-s − 1.56·59-s − 64-s + 3.42·67-s + 4.61·75-s − 4.11·81-s + 3.81·89-s + 2.03·97-s − 3.61·99-s − 2·100-s + 0.769·108-s − 3.38·113-s + 2.27·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7744\)    =    \(2^{6} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(0.493764\)
Root analytic conductor: \(0.838262\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7744,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.352757152\)
\(L(\frac12)\) \(\approx\) \(1.352757152\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( 1 + p T^{2} \)
11$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−14.59281597732568961589511196673, −13.98065060458594722877994523627, −13.42713150093581572562827763778, −12.93565181824292994602284841056, −12.92579629589378485566237522160, −11.96622973037668861103954512623, −10.91767721294012233313446745730, −10.57393021547313230304092123071, −9.631619446080502702009717870228, −9.498041248497130226939725849074, −8.592671388844004873937868322622, −8.542935271710114583439470800606, −7.80406696230682646988448860013, −7.60045353631728464491979491176, −6.39238033606490238102936028054, −5.24350195738179276103553168350, −4.80542217079088952683761723273, −3.59112359359624056066828230442, −3.09544643928030014290032160441, −2.32227705759016695952063673262, 2.32227705759016695952063673262, 3.09544643928030014290032160441, 3.59112359359624056066828230442, 4.80542217079088952683761723273, 5.24350195738179276103553168350, 6.39238033606490238102936028054, 7.60045353631728464491979491176, 7.80406696230682646988448860013, 8.542935271710114583439470800606, 8.592671388844004873937868322622, 9.498041248497130226939725849074, 9.631619446080502702009717870228, 10.57393021547313230304092123071, 10.91767721294012233313446745730, 11.96622973037668861103954512623, 12.92579629589378485566237522160, 12.93565181824292994602284841056, 13.42713150093581572562827763778, 13.98065060458594722877994523627, 14.59281597732568961589511196673

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