Properties

Label 4-4598e2-1.1-c1e2-0-1
Degree $4$
Conductor $21141604$
Sign $1$
Analytic cond. $1348.00$
Root an. cond. $6.05930$
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-s + 3·4-s + 3·5-s + 2·6-s + 5·7-s − 4·8-s − 6·10-s − 3·12-s − 7·13-s − 10·14-s − 3·15-s + 5·16-s + 6·17-s − 2·19-s + 9·20-s − 5·21-s + 6·23-s + 4·24-s + 2·25-s + 14·26-s − 2·27-s + 15·28-s − 9·29-s + 6·30-s − 17·31-s − 6·32-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 3/2·4-s + 1.34·5-s + 0.816·6-s + 1.88·7-s − 1.41·8-s − 1.89·10-s − 0.866·12-s − 1.94·13-s − 2.67·14-s − 0.774·15-s + 5/4·16-s + 1.45·17-s − 0.458·19-s + 2.01·20-s − 1.09·21-s + 1.25·23-s + 0.816·24-s + 2/5·25-s + 2.74·26-s − 0.384·27-s + 2.83·28-s − 1.67·29-s + 1.09·30-s − 3.05·31-s − 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21141604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21141604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21141604\)    =    \(2^{2} \cdot 11^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1348.00\)
Root analytic conductor: \(6.05930\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21141604,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9105820822\)
\(L(\frac12)\) \(\approx\) \(0.9105820822\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 5 T + 15 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 7 T + 33 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 9 T + 73 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 17 T + 129 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 3 T + 79 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + T + 39 T^{2} + p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 5 T + 135 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 9 T + 115 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 10 T + 150 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 2 T + 138 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 3 T + 163 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 18 T + 238 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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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

−8.449995309568280429974547288289, −8.187732300819375626637049702314, −7.64078069173246770004350448486, −7.36863679114542686209646605459, −7.33556763444532057814265401807, −7.05439141241701593415897002875, −6.18488375363243516587109340218, −5.87786408839645781841514074893, −5.64590425191052374461204235225, −5.40129530664191988335826117281, −4.86212121823607138644154734458, −4.73652993625233514690518052521, −3.96844737059657725802438708467, −3.48598730442617386988705840631, −2.67774527791786451728687051170, −2.48757679149046913385337091571, −1.88925969623663867404599253776, −1.58251932017887485058590704752, −1.28722551999193325538959718189, −0.33877803127057278008120094550, 0.33877803127057278008120094550, 1.28722551999193325538959718189, 1.58251932017887485058590704752, 1.88925969623663867404599253776, 2.48757679149046913385337091571, 2.67774527791786451728687051170, 3.48598730442617386988705840631, 3.96844737059657725802438708467, 4.73652993625233514690518052521, 4.86212121823607138644154734458, 5.40129530664191988335826117281, 5.64590425191052374461204235225, 5.87786408839645781841514074893, 6.18488375363243516587109340218, 7.05439141241701593415897002875, 7.33556763444532057814265401807, 7.36863679114542686209646605459, 7.64078069173246770004350448486, 8.187732300819375626637049702314, 8.449995309568280429974547288289

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