Properties

Label 4-666e2-1.1-c1e2-0-33
Degree $4$
Conductor $443556$
Sign $1$
Analytic cond. $28.2815$
Root an. cond. $2.30608$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 5-s + 2·7-s + 4·8-s + 2·10-s + 11-s − 13-s + 4·14-s + 5·16-s + 12·17-s + 4·19-s + 3·20-s + 2·22-s + 3·23-s − 6·25-s − 2·26-s + 6·28-s − 3·29-s + 3·31-s + 6·32-s + 24·34-s + 2·35-s + 2·37-s + 8·38-s + 4·40-s − 9·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.447·5-s + 0.755·7-s + 1.41·8-s + 0.632·10-s + 0.301·11-s − 0.277·13-s + 1.06·14-s + 5/4·16-s + 2.91·17-s + 0.917·19-s + 0.670·20-s + 0.426·22-s + 0.625·23-s − 6/5·25-s − 0.392·26-s + 1.13·28-s − 0.557·29-s + 0.538·31-s + 1.06·32-s + 4.11·34-s + 0.338·35-s + 0.328·37-s + 1.29·38-s + 0.632·40-s − 1.40·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(443556\)    =    \(2^{2} \cdot 3^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(28.2815\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{666} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 443556,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.502485073\)
\(L(\frac12)\) \(\approx\) \(6.502485073\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
37$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - T + 19 T^{2} - p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + T + 23 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 3 T + 19 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_4$ \( 1 + 3 T + 31 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 3 T + 61 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 9 T + 73 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 3 T + 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 11 T + 83 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 21 T + 253 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 7 T + 11 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 20 T + 214 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 4 T + 130 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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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

−10.80173622629491757159567795677, −10.21504624504124819813074099459, −9.941106386784632519402725495147, −9.816211648176531494013209199008, −8.961346271187487164991698361971, −8.522826508070270413003441427636, −7.74028955646227738291857137000, −7.66593324608740923429045774346, −7.28619872223283029369478003157, −6.58726137798134571383683772350, −6.05158282859351272481720035666, −5.61225369719099592638107667636, −5.21146575159579199104920070104, −5.05431496778729029187448796252, −4.12270267488747912083793452666, −3.82255826082229337591182372438, −2.95468931311390124516057995731, −2.89482296470876112563179558494, −1.51849191635781980303548251648, −1.45791910636920151145052736404, 1.45791910636920151145052736404, 1.51849191635781980303548251648, 2.89482296470876112563179558494, 2.95468931311390124516057995731, 3.82255826082229337591182372438, 4.12270267488747912083793452666, 5.05431496778729029187448796252, 5.21146575159579199104920070104, 5.61225369719099592638107667636, 6.05158282859351272481720035666, 6.58726137798134571383683772350, 7.28619872223283029369478003157, 7.66593324608740923429045774346, 7.74028955646227738291857137000, 8.522826508070270413003441427636, 8.961346271187487164991698361971, 9.816211648176531494013209199008, 9.941106386784632519402725495147, 10.21504624504124819813074099459, 10.80173622629491757159567795677

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