Properties

Label 4-9075e2-1.1-c1e2-0-0
Degree $4$
Conductor $82355625$
Sign $1$
Analytic cond. $5251.06$
Root an. cond. $8.51259$
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  − 2·3-s + 4-s + 3·9-s − 2·12-s − 3·16-s − 2·23-s − 4·27-s + 20·31-s + 3·36-s − 14·37-s − 6·47-s + 6·48-s − 9·49-s + 28·53-s − 10·59-s − 7·64-s + 4·67-s + 4·69-s − 26·71-s + 5·81-s + 12·89-s − 2·92-s − 40·93-s − 34·97-s + 8·103-s − 4·108-s + 28·111-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s + 9-s − 0.577·12-s − 3/4·16-s − 0.417·23-s − 0.769·27-s + 3.59·31-s + 1/2·36-s − 2.30·37-s − 0.875·47-s + 0.866·48-s − 9/7·49-s + 3.84·53-s − 1.30·59-s − 7/8·64-s + 0.488·67-s + 0.481·69-s − 3.08·71-s + 5/9·81-s + 1.27·89-s − 0.208·92-s − 4.14·93-s − 3.45·97-s + 0.788·103-s − 0.384·108-s + 2.65·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(82355625\)    =    \(3^{2} \cdot 5^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(5251.06\)
Root analytic conductor: \(8.51259\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9075} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 82355625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.116296900\)
\(L(\frac12)\) \(\approx\) \(1.116296900\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 9 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 33 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 37 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 87 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 17 T + p T^{2} )^{2} \)
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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

−7.76192094165004854603704448548, −7.48222793134634098901671725566, −7.03768230879161747941393746850, −6.75433199910406747599775043338, −6.67912218661966147609319186052, −6.14035006970744507372624204151, −5.92584166136270839402669702447, −5.61959245038821201717991701667, −5.05901342180944699249273902689, −4.87200933415639532965214926366, −4.38047451319579403383181920406, −4.30640883207593010222370834478, −3.70606111693081817478694007497, −3.15767471252882894590347002319, −2.87886768592452048633552488871, −2.35579065082866895125588703332, −1.91674919097966910031818188855, −1.42827924628738642032778457069, −0.948470845475725612766970452003, −0.29700908034329452138770317144, 0.29700908034329452138770317144, 0.948470845475725612766970452003, 1.42827924628738642032778457069, 1.91674919097966910031818188855, 2.35579065082866895125588703332, 2.87886768592452048633552488871, 3.15767471252882894590347002319, 3.70606111693081817478694007497, 4.30640883207593010222370834478, 4.38047451319579403383181920406, 4.87200933415639532965214926366, 5.05901342180944699249273902689, 5.61959245038821201717991701667, 5.92584166136270839402669702447, 6.14035006970744507372624204151, 6.67912218661966147609319186052, 6.75433199910406747599775043338, 7.03768230879161747941393746850, 7.48222793134634098901671725566, 7.76192094165004854603704448548

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