Properties

Label 4-2790e2-1.1-c1e2-0-5
Degree $4$
Conductor $7784100$
Sign $1$
Analytic cond. $496.320$
Root an. cond. $4.71998$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 2·5-s + 6·11-s + 16-s + 2·19-s − 2·20-s − 25-s + 4·29-s − 2·31-s + 20·41-s − 6·44-s + 13·49-s + 12·55-s − 12·59-s − 4·61-s − 64-s + 10·71-s − 2·76-s − 6·79-s + 2·80-s + 2·89-s + 4·95-s + 100-s + 30·101-s − 4·109-s − 4·116-s + 5·121-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.894·5-s + 1.80·11-s + 1/4·16-s + 0.458·19-s − 0.447·20-s − 1/5·25-s + 0.742·29-s − 0.359·31-s + 3.12·41-s − 0.904·44-s + 13/7·49-s + 1.61·55-s − 1.56·59-s − 0.512·61-s − 1/8·64-s + 1.18·71-s − 0.229·76-s − 0.675·79-s + 0.223·80-s + 0.211·89-s + 0.410·95-s + 1/10·100-s + 2.98·101-s − 0.383·109-s − 0.371·116-s + 5/11·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7784100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7784100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7784100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(496.320\)
Root analytic conductor: \(4.71998\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7784100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.745947751\)
\(L(\frac12)\) \(\approx\) \(3.745947751\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
31$C_1$ \( ( 1 + T )^{2} \)
good7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 81 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−9.108393999182225739941639648002, −8.854691586096177477869225921079, −8.253766508185137760681900412881, −7.910757319965529520357772887768, −7.31056034392922677186730411044, −7.28234950793085691563411345893, −6.57578117983521645876269268085, −6.24903876178345709772211330298, −5.97114252062701652524117938244, −5.63780248374158553890614914115, −5.17423726607114961308056746940, −4.59690364956391631141625247499, −4.12131856928668054571521631682, −4.09133533823063571306132515451, −3.33136440373610905353862539951, −2.94097609371198186238399615557, −2.25213487854780927025181149392, −1.83752768207418797841057354295, −1.11379638749635992559441267076, −0.74370061332604406159554930944, 0.74370061332604406159554930944, 1.11379638749635992559441267076, 1.83752768207418797841057354295, 2.25213487854780927025181149392, 2.94097609371198186238399615557, 3.33136440373610905353862539951, 4.09133533823063571306132515451, 4.12131856928668054571521631682, 4.59690364956391631141625247499, 5.17423726607114961308056746940, 5.63780248374158553890614914115, 5.97114252062701652524117938244, 6.24903876178345709772211330298, 6.57578117983521645876269268085, 7.28234950793085691563411345893, 7.31056034392922677186730411044, 7.910757319965529520357772887768, 8.253766508185137760681900412881, 8.854691586096177477869225921079, 9.108393999182225739941639648002

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