Properties

Label 4-5248-1.1-c1e2-0-1
Degree $4$
Conductor $5248$
Sign $1$
Analytic cond. $0.334617$
Root an. cond. $0.760566$
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-s + 4-s + 7-s + 8-s − 5·9-s + 14-s + 16-s − 9·17-s − 5·18-s − 2·23-s + 6·25-s + 28-s + 14·31-s + 32-s − 9·34-s − 5·36-s − 2·41-s − 2·46-s + 47-s − 7·49-s + 6·50-s + 56-s + 14·62-s − 5·63-s + 64-s − 9·68-s − 71-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 5/3·9-s + 0.267·14-s + 1/4·16-s − 2.18·17-s − 1.17·18-s − 0.417·23-s + 6/5·25-s + 0.188·28-s + 2.51·31-s + 0.176·32-s − 1.54·34-s − 5/6·36-s − 0.312·41-s − 0.294·46-s + 0.145·47-s − 49-s + 0.848·50-s + 0.133·56-s + 1.77·62-s − 0.629·63-s + 1/8·64-s − 1.09·68-s − 0.118·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5248\)    =    \(2^{7} \cdot 41\)
Sign: $1$
Analytic conductor: \(0.334617\)
Root analytic conductor: \(0.760566\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5248} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5248,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.151473703\)
\(L(\frac12)\) \(\approx\) \(1.151473703\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
41$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 3 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
53$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 88 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 115 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−12.08471040033142178812298188935, −11.49774760059071294638775733509, −11.34972120287414887663793048521, −10.63983442207465826752676803199, −10.03678681520190696479507866164, −8.985952130047415582147306772252, −8.598549001061339691920262738964, −8.115194846911258697150048109335, −7.09953951617682447790222367306, −6.37739946488882099217809141615, −5.95471463639550373755376584555, −4.87065807995594601501699640400, −4.49349673887596513143168410470, −3.15129202944543872519784358984, −2.37502736514194649668568101270, 2.37502736514194649668568101270, 3.15129202944543872519784358984, 4.49349673887596513143168410470, 4.87065807995594601501699640400, 5.95471463639550373755376584555, 6.37739946488882099217809141615, 7.09953951617682447790222367306, 8.115194846911258697150048109335, 8.598549001061339691920262738964, 8.985952130047415582147306772252, 10.03678681520190696479507866164, 10.63983442207465826752676803199, 11.34972120287414887663793048521, 11.49774760059071294638775733509, 12.08471040033142178812298188935

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