Properties

Label 4-1134e2-1.1-c1e2-0-50
Degree $4$
Conductor $1285956$
Sign $1$
Analytic cond. $81.9936$
Root an. cond. $3.00915$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s + 4·5-s − 2·7-s − 4·8-s − 8·10-s + 2·11-s − 6·13-s + 4·14-s + 5·16-s + 14·17-s − 2·19-s + 12·20-s − 4·22-s + 2·23-s + 5·25-s + 12·26-s − 6·28-s + 10·29-s + 6·31-s − 6·32-s − 28·34-s − 8·35-s + 4·37-s + 4·38-s − 16·40-s + 12·41-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 1.78·5-s − 0.755·7-s − 1.41·8-s − 2.52·10-s + 0.603·11-s − 1.66·13-s + 1.06·14-s + 5/4·16-s + 3.39·17-s − 0.458·19-s + 2.68·20-s − 0.852·22-s + 0.417·23-s + 25-s + 2.35·26-s − 1.13·28-s + 1.85·29-s + 1.07·31-s − 1.06·32-s − 4.80·34-s − 1.35·35-s + 0.657·37-s + 0.648·38-s − 2.52·40-s + 1.87·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1285956\)    =    \(2^{2} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(81.9936\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1285956,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.842703433\)
\(L(\frac12)\) \(\approx\) \(1.842703433\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T - 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 2 T + 92 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 10 T + 156 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 20 T + 230 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 20 T + 243 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 2 T - 76 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 6 T + 139 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.764491109229933810634573918241, −9.695210987152636214352896835533, −9.340457984359283769773850794833, −9.214678306585608311542763007920, −8.289468455539870886670604642057, −8.073173611060736142642857294085, −7.50150875808826303243150444821, −7.39123445470273608523883788881, −6.52320000086489884650777301024, −6.42458775102284963770257394605, −5.86013973546035986635768945154, −5.70335391592220612498138019709, −4.98664153712767081207825785621, −4.56348919810310256435831712296, −3.48244777814721618789283460379, −3.13081215238899425447998646359, −2.37442263818242509803958140776, −2.30127166572978617752789484775, −1.09808247313066073318935895647, −0.953241572069657719294986291341, 0.953241572069657719294986291341, 1.09808247313066073318935895647, 2.30127166572978617752789484775, 2.37442263818242509803958140776, 3.13081215238899425447998646359, 3.48244777814721618789283460379, 4.56348919810310256435831712296, 4.98664153712767081207825785621, 5.70335391592220612498138019709, 5.86013973546035986635768945154, 6.42458775102284963770257394605, 6.52320000086489884650777301024, 7.39123445470273608523883788881, 7.50150875808826303243150444821, 8.073173611060736142642857294085, 8.289468455539870886670604642057, 9.214678306585608311542763007920, 9.340457984359283769773850794833, 9.695210987152636214352896835533, 9.764491109229933810634573918241

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