Properties

Label 4-1815e2-1.1-c0e2-0-1
Degree $4$
Conductor $3294225$
Sign $1$
Analytic cond. $0.820479$
Root an. cond. $0.951736$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4-s − 2·5-s + 3·9-s + 2·12-s − 4·15-s − 2·20-s − 2·23-s + 3·25-s + 4·27-s − 2·31-s + 3·36-s − 6·45-s + 2·47-s + 2·49-s + 2·53-s − 4·60-s − 64-s − 4·69-s + 6·75-s + 5·81-s − 2·92-s − 4·93-s + 3·100-s + 4·108-s − 2·113-s + 4·115-s + ⋯
L(s)  = 1  + 2·3-s + 4-s − 2·5-s + 3·9-s + 2·12-s − 4·15-s − 2·20-s − 2·23-s + 3·25-s + 4·27-s − 2·31-s + 3·36-s − 6·45-s + 2·47-s + 2·49-s + 2·53-s − 4·60-s − 64-s − 4·69-s + 6·75-s + 5·81-s − 2·92-s − 4·93-s + 3·100-s + 4·108-s − 2·113-s + 4·115-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3294225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(0.820479\)
Root analytic conductor: \(0.951736\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1815} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3294225,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.234043244\)
\(L(\frac12)\) \(\approx\) \(2.234043244\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 - T^{2} + T^{4} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2^2$ \( 1 - T^{2} + T^{4} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_2$ \( ( 1 + T + T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 + T + T^{2} )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2$ \( ( 1 - T + T^{2} )^{2} \)
53$C_2$ \( ( 1 - T + T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2^2$ \( 1 - T^{2} + T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2^2$ \( 1 - T^{2} + T^{4} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−9.291526768339868631943004335147, −9.264180899765024003070754522505, −8.599220408249702313754769634163, −8.570255334305873910561405012797, −8.040363013551406591796433611988, −7.61171311947797003367397914238, −7.45545782158029571657243400957, −7.03435184078224158050080459254, −6.97244090172572695914480675444, −6.23789872987372127702708394998, −5.64120301871769792843476013877, −5.07644779908904633346197681159, −4.16578495649104800449671578928, −4.14513960599703584240628432200, −3.90090377276499587792199817408, −3.31538331717361265058665833998, −2.84666865869189123637092785498, −2.25850489647596883471035584227, −2.03033035824921639371144693890, −1.04700972055815919408705427648, 1.04700972055815919408705427648, 2.03033035824921639371144693890, 2.25850489647596883471035584227, 2.84666865869189123637092785498, 3.31538331717361265058665833998, 3.90090377276499587792199817408, 4.14513960599703584240628432200, 4.16578495649104800449671578928, 5.07644779908904633346197681159, 5.64120301871769792843476013877, 6.23789872987372127702708394998, 6.97244090172572695914480675444, 7.03435184078224158050080459254, 7.45545782158029571657243400957, 7.61171311947797003367397914238, 8.040363013551406591796433611988, 8.570255334305873910561405012797, 8.599220408249702313754769634163, 9.264180899765024003070754522505, 9.291526768339868631943004335147

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