Properties

Label 4-3e2-1.1-c10e2-0-0
Degree $4$
Conductor $9$
Sign $1$
Analytic cond. $3.63310$
Root an. cond. $1.38060$
Motivic weight $10$
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  − 54·3-s + 1.32e3·4-s + 3.44e4·7-s − 5.61e4·9-s − 7.17e4·12-s − 3.39e5·13-s + 7.15e5·16-s − 1.89e6·19-s − 1.86e6·21-s + 1.14e7·25-s + 6.21e6·27-s + 4.57e7·28-s − 5.95e7·31-s − 7.45e7·36-s − 1.21e8·37-s + 1.83e7·39-s + 2.14e8·43-s − 3.86e7·48-s + 3.26e8·49-s − 4.50e8·52-s + 1.02e8·57-s + 2.06e9·61-s − 1.93e9·63-s − 4.42e8·64-s + 3.75e9·67-s − 5.69e9·73-s − 6.17e8·75-s + ⋯
L(s)  = 1  − 2/9·3-s + 1.29·4-s + 2.05·7-s − 0.950·9-s − 0.288·12-s − 0.913·13-s + 0.681·16-s − 0.766·19-s − 0.455·21-s + 1.17·25-s + 0.433·27-s + 2.65·28-s − 2.08·31-s − 1.23·36-s − 1.75·37-s + 0.203·39-s + 1.46·43-s − 0.151·48-s + 1.15·49-s − 1.18·52-s + 0.170·57-s + 2.44·61-s − 1.94·63-s − 0.412·64-s + 2.78·67-s − 2.74·73-s − 0.260·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $1$
Analytic conductor: \(3.63310\)
Root analytic conductor: \(1.38060\)
Motivic weight: \(10\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9,\ (\ :5, 5),\ 1)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.797533869\)
\(L(\frac12)\) \(\approx\) \(1.797533869\)
\(L(6)\) 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$C_2$ \( 1 + 2 p^{3} T + p^{10} T^{2} \)
good2$C_2^2$ \( 1 - 83 p^{4} T^{2} + p^{20} T^{4} \)
5$C_2^2$ \( 1 - 2288266 p T^{2} + p^{20} T^{4} \)
7$C_2$ \( ( 1 - 2462 p T + p^{10} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 16976849522 T^{2} + p^{20} T^{4} \)
13$C_2$ \( ( 1 + 169654 T + p^{10} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 3914170410818 T^{2} + p^{20} T^{4} \)
19$C_2$ \( ( 1 + 949462 T + p^{10} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 75790028393378 T^{2} + p^{20} T^{4} \)
29$C_2^2$ \( 1 - 831288000078482 T^{2} + p^{20} T^{4} \)
31$C_2$ \( ( 1 + 29793118 T + p^{10} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 60811846 T + p^{10} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 6157996032931678 T^{2} + p^{20} T^{4} \)
43$C_2$ \( ( 1 - 107419706 T + p^{10} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 33376598707262018 T^{2} + p^{20} T^{4} \)
53$C_2^2$ \( 1 - 312876100791567218 T^{2} + p^{20} T^{4} \)
59$C_2^2$ \( 1 - 600827685707033522 T^{2} + p^{20} T^{4} \)
61$C_2$ \( ( 1 - 1030793642 T + p^{10} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 1876742474 T + p^{10} T^{2} )^{2} \)
71$C_2^2$ \( 1 + 690030731290713118 T^{2} + p^{20} T^{4} \)
73$C_2$ \( ( 1 + 2846528494 T + p^{10} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 1488647618 T + p^{10} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 29429698146400299218 T^{2} + p^{20} T^{4} \)
89$C_2^2$ \( 1 - 26116812713945754722 T^{2} + p^{20} T^{4} \)
97$C_2$ \( ( 1 + 1592948926 T + p^{10} T^{2} )^{2} \)
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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

−24.84457899962566353199445389259, −24.20318576127480771016158811837, −23.74338877657051435643787226644, −22.44011139079520987686395056833, −21.47064512361908584399553012875, −20.58561056656826117159746637880, −20.28532374838293855514717649071, −19.08117028631012313255617425684, −17.70115337145780505421991111457, −17.18075787907553319914893773469, −16.11038654296428834026969403329, −14.67416678267482593441362433697, −14.52431261187424724044323929269, −12.36693015680932163830038383401, −11.29941217197126212856117573402, −10.85759080033753171555976318191, −8.499579027723398985905653048973, −7.16165388174068795820950336271, −5.24946800564837304581190225336, −2.08462692934217493550687428847, 2.08462692934217493550687428847, 5.24946800564837304581190225336, 7.16165388174068795820950336271, 8.499579027723398985905653048973, 10.85759080033753171555976318191, 11.29941217197126212856117573402, 12.36693015680932163830038383401, 14.52431261187424724044323929269, 14.67416678267482593441362433697, 16.11038654296428834026969403329, 17.18075787907553319914893773469, 17.70115337145780505421991111457, 19.08117028631012313255617425684, 20.28532374838293855514717649071, 20.58561056656826117159746637880, 21.47064512361908584399553012875, 22.44011139079520987686395056833, 23.74338877657051435643787226644, 24.20318576127480771016158811837, 24.84457899962566353199445389259

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