Properties

Label 4-1492992-1.1-c1e2-0-2
Degree $4$
Conductor $1492992$
Sign $1$
Analytic cond. $95.1944$
Root an. cond. $3.12358$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 4·11-s + 6·17-s + 4·19-s − 3·25-s + 27-s − 4·33-s − 2·41-s − 10·43-s − 49-s + 6·51-s + 4·57-s + 4·59-s + 17·67-s − 9·73-s − 3·75-s + 81-s + 6·83-s − 16·89-s + 14·97-s − 4·99-s + 16·107-s − 20·113-s − 10·121-s − 2·123-s + 127-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.20·11-s + 1.45·17-s + 0.917·19-s − 3/5·25-s + 0.192·27-s − 0.696·33-s − 0.312·41-s − 1.52·43-s − 1/7·49-s + 0.840·51-s + 0.529·57-s + 0.520·59-s + 2.07·67-s − 1.05·73-s − 0.346·75-s + 1/9·81-s + 0.658·83-s − 1.69·89-s + 1.42·97-s − 0.402·99-s + 1.54·107-s − 1.88·113-s − 0.909·121-s − 0.180·123-s + 0.0887·127-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1492992,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.404475735\)
\(L(\frac12)\) \(\approx\) \(2.404475735\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
good5$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 39 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 77 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 45 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \)
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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

−7.85728262763095169313843948937, −7.68861371188996232540470309728, −7.10307071118499378347990234969, −6.76577012602044804816060945872, −6.15227370711194534371797717052, −5.57144035110940339483931759948, −5.28691017079695308551340771481, −4.97029458115575190509821682229, −4.23572693451180686189833399558, −3.72484491861137524032454209310, −3.13656584290078523571036422082, −2.95100817360623298872957202158, −2.13549144191685407044870532758, −1.57256997604759351981182987109, −0.64887378648367066807085322574, 0.64887378648367066807085322574, 1.57256997604759351981182987109, 2.13549144191685407044870532758, 2.95100817360623298872957202158, 3.13656584290078523571036422082, 3.72484491861137524032454209310, 4.23572693451180686189833399558, 4.97029458115575190509821682229, 5.28691017079695308551340771481, 5.57144035110940339483931759948, 6.15227370711194534371797717052, 6.76577012602044804816060945872, 7.10307071118499378347990234969, 7.68861371188996232540470309728, 7.85728262763095169313843948937

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