Properties

Label 4-1815e2-1.1-c1e2-0-21
Degree $4$
Conductor $3294225$
Sign $1$
Analytic cond. $210.042$
Root an. cond. $3.80694$
Motivic weight $1$
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-s − 2·3-s − 2·4-s − 2·5-s − 2·6-s + 2·7-s − 3·8-s + 3·9-s − 2·10-s + 4·12-s + 6·13-s + 2·14-s + 4·15-s + 16-s − 2·17-s + 3·18-s + 4·19-s + 4·20-s − 4·21-s + 4·23-s + 6·24-s + 3·25-s + 6·26-s − 4·27-s − 4·28-s + 2·29-s + 4·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 4-s − 0.894·5-s − 0.816·6-s + 0.755·7-s − 1.06·8-s + 9-s − 0.632·10-s + 1.15·12-s + 1.66·13-s + 0.534·14-s + 1.03·15-s + 1/4·16-s − 0.485·17-s + 0.707·18-s + 0.917·19-s + 0.894·20-s − 0.872·21-s + 0.834·23-s + 1.22·24-s + 3/5·25-s + 1.17·26-s − 0.769·27-s − 0.755·28-s + 0.371·29-s + 0.730·30-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: induced by $\chi_{1815} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3294225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.893685617\)
\(L(\frac12)\) \(\approx\) \(1.893685617\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_4$ \( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 6 T + 90 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 53 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 6 T + 111 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 18 T + 210 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 12 T + 162 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 12 T + 149 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 14 T + 222 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 30 T + 414 T^{2} - 30 p T^{3} + p^{2} T^{4} \)
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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

−9.325012064678932692986700752254, −9.037052855310290616352642500388, −8.582099751421481240773285062363, −8.460909499670961194537051937551, −7.79315117774579275858784669052, −7.47676152222131388133449886600, −7.13298187166834420041071977118, −6.50230302422962949453235552369, −6.04276253560562332359592948124, −5.87906373830787910810902376787, −5.19520931276398807688658271499, −5.00708051626643676653931728210, −4.48527348226146045173814460850, −4.30874286982828647610998319217, −3.70187628979622433491499798442, −3.48742788429213568835127407996, −2.74253557029201212677306955913, −1.83600078314410590847902660307, −0.830149055478123674746407958472, −0.78651131815844179878191431538, 0.78651131815844179878191431538, 0.830149055478123674746407958472, 1.83600078314410590847902660307, 2.74253557029201212677306955913, 3.48742788429213568835127407996, 3.70187628979622433491499798442, 4.30874286982828647610998319217, 4.48527348226146045173814460850, 5.00708051626643676653931728210, 5.19520931276398807688658271499, 5.87906373830787910810902376787, 6.04276253560562332359592948124, 6.50230302422962949453235552369, 7.13298187166834420041071977118, 7.47676152222131388133449886600, 7.79315117774579275858784669052, 8.460909499670961194537051937551, 8.582099751421481240773285062363, 9.037052855310290616352642500388, 9.325012064678932692986700752254

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