Properties

Label 4-48e4-1.1-c2e2-0-2
Degree $4$
Conductor $5308416$
Sign $1$
Analytic cond. $3941.25$
Root an. cond. $7.92334$
Motivic weight $2$
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  − 32·17-s − 14·25-s − 160·41-s + 98·49-s − 220·73-s + 320·89-s − 260·97-s + 448·113-s − 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 238·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 1.88·17-s − 0.559·25-s − 3.90·41-s + 2·49-s − 3.01·73-s + 3.59·89-s − 2.68·97-s + 3.96·113-s − 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.40·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(3941.25\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2304} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5308416,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1838476365\)
\(L(\frac12)\) \(\approx\) \(0.1838476365\)
\(L(2)\) 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 \( 1 \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 6 T + p^{2} T^{2} )( 1 + 6 T + p^{2} T^{2} ) \)
7$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
11$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 24 T + p^{2} T^{2} )( 1 + 24 T + p^{2} T^{2} ) \)
17$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{2} \)
19$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
29$C_2$ \( ( 1 - 42 T + p^{2} T^{2} )( 1 + 42 T + p^{2} T^{2} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
37$C_2$ \( ( 1 - 24 T + p^{2} T^{2} )( 1 + 24 T + p^{2} T^{2} ) \)
41$C_2$ \( ( 1 + 80 T + p^{2} T^{2} )^{2} \)
43$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
53$C_2$ \( ( 1 - 90 T + p^{2} T^{2} )( 1 + 90 T + p^{2} T^{2} ) \)
59$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 120 T + p^{2} T^{2} )( 1 + 120 T + p^{2} T^{2} ) \)
67$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2$ \( ( 1 + 110 T + p^{2} T^{2} )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
83$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
89$C_2$ \( ( 1 - 160 T + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 130 T + p^{2} T^{2} )^{2} \)
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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

−9.232209136851508745226665343860, −8.663925471996682732216225294959, −8.410717922423940163256048948958, −7.85270983954925281257458878700, −7.46170853131590429054819953335, −7.00292305791717329107484390345, −6.73710319460723332686557801606, −6.40632977768699523168355530942, −5.87291774803100614173443823406, −5.50588378388715542281210878335, −5.01722402598296722380591871320, −4.52149573289662922536254066216, −4.36135707365332285886039434454, −3.52158219445314296362992092321, −3.48727403363898510937007883629, −2.65013416968999020829653281068, −2.20624476668426273428652406136, −1.77237310731648065597035701029, −1.12192597085330551879615934889, −0.10680510751916947564980557105, 0.10680510751916947564980557105, 1.12192597085330551879615934889, 1.77237310731648065597035701029, 2.20624476668426273428652406136, 2.65013416968999020829653281068, 3.48727403363898510937007883629, 3.52158219445314296362992092321, 4.36135707365332285886039434454, 4.52149573289662922536254066216, 5.01722402598296722380591871320, 5.50588378388715542281210878335, 5.87291774803100614173443823406, 6.40632977768699523168355530942, 6.73710319460723332686557801606, 7.00292305791717329107484390345, 7.46170853131590429054819953335, 7.85270983954925281257458878700, 8.410717922423940163256048948958, 8.663925471996682732216225294959, 9.232209136851508745226665343860

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