Properties

Label 4-1492992-1.1-c1e2-0-6
Degree $4$
Conductor $1492992$
Sign $1$
Analytic cond. $95.1944$
Root an. cond. $3.12358$
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  − 4·11-s + 12·17-s + 10·19-s − 6·25-s + 16·41-s + 8·43-s − 13·49-s + 28·59-s + 26·67-s + 18·73-s − 24·83-s − 4·89-s + 2·97-s + 4·107-s − 20·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 25·169-s + 173-s + ⋯
L(s)  = 1  − 1.20·11-s + 2.91·17-s + 2.29·19-s − 6/5·25-s + 2.49·41-s + 1.21·43-s − 1.85·49-s + 3.64·59-s + 3.17·67-s + 2.10·73-s − 2.63·83-s − 0.423·89-s + 0.203·97-s + 0.386·107-s − 1.88·113-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.92·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1492992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1492992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1492992\)    =    \(2^{11} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(95.1944\)
Root analytic conductor: \(3.12358\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1492992} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1492992,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.597918890\)
\(L(\frac12)\) \(\approx\) \(2.597918890\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
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

−7.86749870783756385045860969127, −7.57195972780640499908950962067, −7.24764130899628727569423119197, −6.71110929993388671275045148454, −5.92718935464768987920858473492, −5.61589452328752483822361023345, −5.29554848587607308034135507932, −5.17169676997140923385106697279, −4.23592786371004028357777676518, −3.58529057716851715664298339034, −3.47453539619149025125996376504, −2.66421157652505017015711163951, −2.35447010108907921399113958644, −1.19497678164755144713673653335, −0.826140575077611309933401896495, 0.826140575077611309933401896495, 1.19497678164755144713673653335, 2.35447010108907921399113958644, 2.66421157652505017015711163951, 3.47453539619149025125996376504, 3.58529057716851715664298339034, 4.23592786371004028357777676518, 5.17169676997140923385106697279, 5.29554848587607308034135507932, 5.61589452328752483822361023345, 5.92718935464768987920858473492, 6.71110929993388671275045148454, 7.24764130899628727569423119197, 7.57195972780640499908950962067, 7.86749870783756385045860969127

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