Properties

Label 4-1815e2-1.1-c1e2-0-0
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 − 2·4-s + 3·5-s + 9-s + 4·12-s − 6·15-s − 6·20-s − 3·23-s + 4·25-s + 4·27-s − 2·31-s − 2·36-s − 18·37-s + 3·45-s − 21·47-s + 2·49-s + 18·53-s − 15·59-s + 12·60-s + 8·64-s + 6·69-s − 21·71-s − 8·75-s − 11·81-s − 9·89-s + 6·92-s + 4·93-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s + 1.34·5-s + 1/3·9-s + 1.15·12-s − 1.54·15-s − 1.34·20-s − 0.625·23-s + 4/5·25-s + 0.769·27-s − 0.359·31-s − 1/3·36-s − 2.95·37-s + 0.447·45-s − 3.06·47-s + 2/7·49-s + 2.47·53-s − 1.95·59-s + 1.54·60-s + 64-s + 0.722·69-s − 2.49·71-s − 0.923·75-s − 1.22·81-s − 0.953·89-s + 0.625·92-s + 0.414·93-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

\(L(1)\) \(\approx\) \(0.3248454636\)
\(L(\frac12)\) \(\approx\) \(0.3248454636\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 3 T + p T^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 + 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 88 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \)
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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

−7.30824845984976924238990703509, −6.93275094325729312340701527268, −6.62999563558761094718762867470, −6.15600627177212845264506692506, −5.69123742621604292966783911979, −5.53185949754034281137058368951, −4.98374361647872624899648170312, −4.80888308439694567367942689707, −4.24491109938982553301303420029, −3.64729513921333045933162350227, −3.14670017183830822086441793621, −2.46331553769077827732073358538, −1.73580582125540547611989809817, −1.39383683977412058707685019539, −0.23377236844491829273595134830, 0.23377236844491829273595134830, 1.39383683977412058707685019539, 1.73580582125540547611989809817, 2.46331553769077827732073358538, 3.14670017183830822086441793621, 3.64729513921333045933162350227, 4.24491109938982553301303420029, 4.80888308439694567367942689707, 4.98374361647872624899648170312, 5.53185949754034281137058368951, 5.69123742621604292966783911979, 6.15600627177212845264506692506, 6.62999563558761094718762867470, 6.93275094325729312340701527268, 7.30824845984976924238990703509

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