Properties

Label 4-1815e2-1.1-c1e2-0-3
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 + 9-s − 2·12-s − 3·16-s − 6·23-s + 25-s − 4·27-s + 4·31-s − 36-s + 6·37-s − 12·47-s − 6·48-s − 6·49-s − 12·53-s + 6·59-s + 7·64-s + 10·67-s − 12·69-s − 6·71-s + 2·75-s − 11·81-s − 12·89-s + 6·92-s + 8·93-s + 16·97-s − 100-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s + 1/3·9-s − 0.577·12-s − 3/4·16-s − 1.25·23-s + 1/5·25-s − 0.769·27-s + 0.718·31-s − 1/6·36-s + 0.986·37-s − 1.75·47-s − 0.866·48-s − 6/7·49-s − 1.64·53-s + 0.781·59-s + 7/8·64-s + 1.22·67-s − 1.44·69-s − 0.712·71-s + 0.230·75-s − 1.22·81-s − 1.27·89-s + 0.625·92-s + 0.829·93-s + 1.62·97-s − 0.0999·100-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

\(L(1)\) \(\approx\) \(1.891340900\)
\(L(\frac12)\) \(\approx\) \(1.891340900\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
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$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 84 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 60 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 118 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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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

−7.69618973934914551560896261224, −7.10163113987934299291541895546, −6.80070768556132965159601654305, −6.22253269685592393321241841059, −5.96576654652179918024248534366, −5.37770263702229360830338463508, −4.76009195600617304904243410496, −4.53329461632506939934285332302, −4.05478741424434383969359488893, −3.48884619719388812095500287802, −3.13312794820363966044532086202, −2.58039274511799943505008285886, −2.04673585624774525925522285066, −1.55626959829380241786853714115, −0.46274039392363101096520883657, 0.46274039392363101096520883657, 1.55626959829380241786853714115, 2.04673585624774525925522285066, 2.58039274511799943505008285886, 3.13312794820363966044532086202, 3.48884619719388812095500287802, 4.05478741424434383969359488893, 4.53329461632506939934285332302, 4.76009195600617304904243410496, 5.37770263702229360830338463508, 5.96576654652179918024248534366, 6.22253269685592393321241841059, 6.80070768556132965159601654305, 7.10163113987934299291541895546, 7.69618973934914551560896261224

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