Properties

Label 4-48e4-1.1-c2e2-0-4
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  − 20·13-s − 48·25-s − 48·37-s − 98·49-s + 240·61-s − 192·73-s + 288·97-s − 364·109-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 38·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 1.53·13-s − 1.91·25-s − 1.29·37-s − 2·49-s + 3.93·61-s − 2.63·73-s + 2.96·97-s − 3.33·109-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 − 0.224·169-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5308416,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2675568326\)
\(L(\frac12)\) \(\approx\) \(0.2675568326\)
\(L(2)\) 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$C_2^2$ \( 1 + 48 T^{2} + p^{4} T^{4} \)
7$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
13$C_2$ \( ( 1 + 10 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 480 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
29$C_2^2$ \( 1 + 1680 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 24 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 1440 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
53$C_2^2$ \( 1 - 5040 T^{2} + p^{4} T^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
61$C_2$ \( ( 1 - 120 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2$ \( ( 1 + 96 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
89$C_2^2$ \( 1 + 12480 T^{2} + p^{4} T^{4} \)
97$C_2$ \( ( 1 - 144 T + p^{2} T^{2} )^{2} \)
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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

−9.695855784929875814110270719981, −8.564634579849044980618849756050, −8.258583889709116907322130903071, −7.81168594740767994173515429461, −7.55735621592454448647834806651, −6.97767996621339476319185480452, −6.87972779688511387690117164584, −6.35693005428886214399257719511, −5.67994516340178108389434591084, −5.59764050585310530830430213612, −5.06894533191213431218411588291, −4.58851345151209340250416713555, −4.30290249658858864135945210930, −3.54857571936561452205448044510, −3.43665805238807338015252311561, −2.66085326806900251101363509893, −2.15601233509950635000065410369, −1.86975634581687124417112877302, −1.06217249926952351275689743864, −0.13208250704944538916571866129, 0.13208250704944538916571866129, 1.06217249926952351275689743864, 1.86975634581687124417112877302, 2.15601233509950635000065410369, 2.66085326806900251101363509893, 3.43665805238807338015252311561, 3.54857571936561452205448044510, 4.30290249658858864135945210930, 4.58851345151209340250416713555, 5.06894533191213431218411588291, 5.59764050585310530830430213612, 5.67994516340178108389434591084, 6.35693005428886214399257719511, 6.87972779688511387690117164584, 6.97767996621339476319185480452, 7.55735621592454448647834806651, 7.81168594740767994173515429461, 8.258583889709116907322130903071, 8.564634579849044980618849756050, 9.695855784929875814110270719981

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