Properties

Label 4-1815e2-1.1-c0e2-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(1.253249032\)
\(L(\frac12)\) \(\approx\) \(1.253249032\)
\(L(1)\) 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

−9.656159348438347892927595392056, −9.537931299313523383273877423407, −8.950305856341564658739283209598, −8.778033178593796715735826706120, −7.84648844931249540853838992816, −7.23610872739760395777815749759, −7.20327146837067514980066063956, −6.69962481250731999581600669534, −6.42592339820266726956159402619, −6.06243175712680702213385968073, −5.72484151294612826459912536393, −5.30067903487659601179785626884, −4.89973313744619903837494091082, −4.76751469542819473074050634909, −3.90850182961347727927371126908, −3.20946499275907622107630563419, −2.66212566692108012495035884463, −1.86020521558123107755363071851, −1.70759946372457092048555556385, −1.02019953795764666445398560065, 1.02019953795764666445398560065, 1.70759946372457092048555556385, 1.86020521558123107755363071851, 2.66212566692108012495035884463, 3.20946499275907622107630563419, 3.90850182961347727927371126908, 4.76751469542819473074050634909, 4.89973313744619903837494091082, 5.30067903487659601179785626884, 5.72484151294612826459912536393, 6.06243175712680702213385968073, 6.42592339820266726956159402619, 6.69962481250731999581600669534, 7.20327146837067514980066063956, 7.23610872739760395777815749759, 7.84648844931249540853838992816, 8.778033178593796715735826706120, 8.950305856341564658739283209598, 9.537931299313523383273877423407, 9.656159348438347892927595392056

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