Properties

Label 4-285e2-1.1-c1e2-0-8
Degree $4$
Conductor $81225$
Sign $1$
Analytic cond. $5.17897$
Root an. cond. $1.50855$
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  + 2·2-s + 2·3-s + 4-s − 2·5-s + 4·6-s + 3·9-s − 4·10-s + 4·11-s + 2·12-s − 4·13-s − 4·15-s + 16-s + 8·17-s + 6·18-s − 2·19-s − 2·20-s + 8·22-s + 4·23-s + 3·25-s − 8·26-s + 4·27-s − 8·30-s − 12·31-s − 2·32-s + 8·33-s + 16·34-s + 3·36-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s + 1.63·6-s + 9-s − 1.26·10-s + 1.20·11-s + 0.577·12-s − 1.10·13-s − 1.03·15-s + 1/4·16-s + 1.94·17-s + 1.41·18-s − 0.458·19-s − 0.447·20-s + 1.70·22-s + 0.834·23-s + 3/5·25-s − 1.56·26-s + 0.769·27-s − 1.46·30-s − 2.15·31-s − 0.353·32-s + 1.39·33-s + 2.74·34-s + 1/2·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(81225\)    =    \(3^{2} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(5.17897\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{285} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 81225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.835754495\)
\(L(\frac12)\) \(\approx\) \(3.835754495\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 76 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 8 T + 80 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 16 T + 132 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 10 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 16 T + 198 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 20 T + 258 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 8 T + 96 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 28 T + 372 T^{2} + 28 p T^{3} + p^{2} T^{4} \)
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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

−12.23511320145276539552922414293, −11.97617666403385932581912751005, −11.21411063461569254226128164985, −10.78753297735059819187015212485, −10.19067810920104136365419417778, −9.600468563980552849636352847966, −9.061853962899491974237723894976, −8.927516763719285408940852117938, −8.071291712576376438194928429427, −7.54003299218301302356581633802, −7.32162446601290164924871155949, −6.83378106786382026747386148724, −5.74500436770186020896138892320, −5.49238882760750975616135169751, −4.56926424233502251010085086314, −4.32766183425900849295621201115, −3.60433233639417382193935484750, −3.40287857323995405053371516534, −2.53627796200111631637271098706, −1.38227551095877088266721583956, 1.38227551095877088266721583956, 2.53627796200111631637271098706, 3.40287857323995405053371516534, 3.60433233639417382193935484750, 4.32766183425900849295621201115, 4.56926424233502251010085086314, 5.49238882760750975616135169751, 5.74500436770186020896138892320, 6.83378106786382026747386148724, 7.32162446601290164924871155949, 7.54003299218301302356581633802, 8.071291712576376438194928429427, 8.927516763719285408940852117938, 9.061853962899491974237723894976, 9.600468563980552849636352847966, 10.19067810920104136365419417778, 10.78753297735059819187015212485, 11.21411063461569254226128164985, 11.97617666403385932581912751005, 12.23511320145276539552922414293

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