Properties

Degree $4$
Conductor $49787136$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 4·11-s + 4·17-s − 8·19-s + 4·23-s + 4·25-s − 16·31-s − 8·37-s + 4·41-s + 16·43-s + 8·47-s + 4·53-s − 16·55-s + 16·59-s + 8·61-s + 16·67-s + 4·71-s − 8·73-s + 16·79-s + 8·83-s − 16·85-s − 4·89-s + 32·95-s − 24·97-s − 4·101-s − 16·103-s + 12·107-s + ⋯
L(s)  = 1  − 1.78·5-s + 1.20·11-s + 0.970·17-s − 1.83·19-s + 0.834·23-s + 4/5·25-s − 2.87·31-s − 1.31·37-s + 0.624·41-s + 2.43·43-s + 1.16·47-s + 0.549·53-s − 2.15·55-s + 2.08·59-s + 1.02·61-s + 1.95·67-s + 0.474·71-s − 0.936·73-s + 1.80·79-s + 0.878·83-s − 1.73·85-s − 0.423·89-s + 3.28·95-s − 2.43·97-s − 0.398·101-s − 1.57·103-s + 1.16·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(49787136\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{7056} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 49787136,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.031725438\)
\(L(\frac12)\) \(\approx\) \(2.031725438\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$C_4$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 16 T + 118 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 4 T + 84 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 8 T + 112 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 24 T + 320 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
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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

−7.999702192274728680272073971249, −7.84559382173864157893131293476, −7.27535810676448797876926330767, −7.22200613144326167171411357154, −6.75133724561195384721406551465, −6.65237302822365979181137794291, −5.85669023914851587293300073168, −5.74235152952742374041253616764, −5.21169154175944496043081488782, −5.04193553406866687484500234067, −4.13978158786338060007048669129, −4.01660104392668705666280523062, −3.82652219361121838120568632629, −3.79100253654556548784504160977, −3.00992753234560890334967470473, −2.53774985939799479818192528144, −2.00266703295827614311349636913, −1.59225477029904112362593146841, −0.68408346635680756096651519106, −0.54688236611422705431268569970, 0.54688236611422705431268569970, 0.68408346635680756096651519106, 1.59225477029904112362593146841, 2.00266703295827614311349636913, 2.53774985939799479818192528144, 3.00992753234560890334967470473, 3.79100253654556548784504160977, 3.82652219361121838120568632629, 4.01660104392668705666280523062, 4.13978158786338060007048669129, 5.04193553406866687484500234067, 5.21169154175944496043081488782, 5.74235152952742374041253616764, 5.85669023914851587293300073168, 6.65237302822365979181137794291, 6.75133724561195384721406551465, 7.22200613144326167171411357154, 7.27535810676448797876926330767, 7.84559382173864157893131293476, 7.999702192274728680272073971249

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