Properties

Label 4-1815e2-1.1-c1e2-0-5
Degree $4$
Conductor $3294225$
Sign $1$
Analytic cond. $210.042$
Root an. cond. $3.80694$
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  − 2·3-s + 4-s + 2·5-s + 3·9-s − 2·12-s − 4·15-s − 3·16-s + 2·20-s + 4·23-s + 3·25-s − 4·27-s − 4·31-s + 3·36-s − 8·37-s + 6·45-s − 12·47-s + 6·48-s + 6·49-s − 8·53-s − 4·59-s − 4·60-s − 7·64-s + 16·67-s − 8·69-s + 16·71-s − 6·75-s − 6·80-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s + 0.894·5-s + 9-s − 0.577·12-s − 1.03·15-s − 3/4·16-s + 0.447·20-s + 0.834·23-s + 3/5·25-s − 0.769·27-s − 0.718·31-s + 1/2·36-s − 1.31·37-s + 0.894·45-s − 1.75·47-s + 0.866·48-s + 6/7·49-s − 1.09·53-s − 0.520·59-s − 0.516·60-s − 7/8·64-s + 1.95·67-s − 0.963·69-s + 1.89·71-s − 0.692·75-s − 0.670·80-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(3294225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(210.042\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3294225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3294225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.583278428\)
\(L(\frac12)\) \(\approx\) \(1.583278428\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−7.23815392841040708057024452919, −6.93369534054451136237238298192, −6.66997089871483937530094116207, −6.33441546971782071295643763649, −5.87511436039108526070458178589, −5.35744616282653305288898933553, −5.14595033301132747372216851028, −4.75081564435104570741968555343, −4.18646610451943193090708612222, −3.56281303427106806080518253523, −3.07807572238901504119232779846, −2.38889252462389521168499293084, −1.86247893551332268542445079595, −1.42566135957896737632755522849, −0.50211099024170605810363493885, 0.50211099024170605810363493885, 1.42566135957896737632755522849, 1.86247893551332268542445079595, 2.38889252462389521168499293084, 3.07807572238901504119232779846, 3.56281303427106806080518253523, 4.18646610451943193090708612222, 4.75081564435104570741968555343, 5.14595033301132747372216851028, 5.35744616282653305288898933553, 5.87511436039108526070458178589, 6.33441546971782071295643763649, 6.66997089871483937530094116207, 6.93369534054451136237238298192, 7.23815392841040708057024452919

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