Properties

Label 4-338e2-1.1-c3e2-0-4
Degree $4$
Conductor $114244$
Sign $1$
Analytic cond. $397.709$
Root an. cond. $4.46571$
Motivic weight $3$
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 − 4·3-s − 36·5-s + 8·6-s − 20·7-s + 8·8-s + 27·9-s + 72·10-s + 48·11-s + 40·14-s + 144·15-s − 16·16-s − 66·17-s − 54·18-s + 16·19-s + 80·21-s − 96·22-s − 168·23-s − 32·24-s + 722·25-s − 260·27-s − 6·29-s − 288·30-s + 40·31-s − 192·33-s + 132·34-s + 720·35-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.769·3-s − 3.21·5-s + 0.544·6-s − 1.07·7-s + 0.353·8-s + 9-s + 2.27·10-s + 1.31·11-s + 0.763·14-s + 2.47·15-s − 1/4·16-s − 0.941·17-s − 0.707·18-s + 0.193·19-s + 0.831·21-s − 0.930·22-s − 1.52·23-s − 0.272·24-s + 5.77·25-s − 1.85·27-s − 0.0384·29-s − 1.75·30-s + 0.231·31-s − 1.01·33-s + 0.665·34-s + 3.47·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114244 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(114244\)    =    \(2^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(397.709\)
Root analytic conductor: \(4.46571\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 114244,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3542792865\)
\(L(\frac12)\) \(\approx\) \(0.3542792865\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p T + p^{2} T^{2} \)
13 \( 1 \)
good3$C_2^2$ \( 1 + 4 T - 11 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \)
5$C_2$ \( ( 1 + 18 T + p^{3} T^{2} )^{2} \)
7$C_2$ \( ( 1 - 17 T + p^{3} T^{2} )( 1 + 37 T + p^{3} T^{2} ) \)
11$C_2^2$ \( 1 - 48 T + 973 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2^2$ \( 1 + 66 T - 557 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 16 T - 6603 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 168 T + 16057 T^{2} + 168 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 + 6 T - 24353 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2$ \( ( 1 - 20 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 + 254 T + 13863 T^{2} + 254 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2^2$ \( 1 - 390 T + 83179 T^{2} - 390 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 - 124 T - 64131 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \)
47$C_2$ \( ( 1 + 468 T + p^{3} T^{2} )^{2} \)
53$C_2$ \( ( 1 - 558 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 96 T - 196163 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 826 T + 455295 T^{2} - 826 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 160 T - 275163 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2^2$ \( 1 - 420 T - 181511 T^{2} - 420 p^{3} T^{3} + p^{6} T^{4} \)
73$C_2$ \( ( 1 - 362 T + p^{3} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 776 T + p^{3} T^{2} )^{2} \)
83$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 1626 T + 1938907 T^{2} + 1626 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2^2$ \( 1 - 1294 T + 761763 T^{2} - 1294 p^{3} T^{3} + p^{6} T^{4} \)
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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

−11.27437239494547146492453416813, −11.21782759606127006318309868009, −10.44819065425930979156485156588, −9.973648641765421886781701531210, −9.491872697363616112831980572770, −8.976558111229072345918055801937, −8.382821401789467171793381100729, −8.102842478582627260086737971183, −7.52904831264959363218795114306, −7.10231972777536168992340381319, −6.77342476693903807043098489924, −6.29902244239417238411711653275, −5.39890127734916384538248571798, −4.50069135023256668013015384362, −4.17526339642412635099799215653, −3.62268077772743901137959722049, −3.58922201192660747473123659914, −2.07666952277646208466002238400, −0.69880204178286530275223249204, −0.43769635512455103228218936551, 0.43769635512455103228218936551, 0.69880204178286530275223249204, 2.07666952277646208466002238400, 3.58922201192660747473123659914, 3.62268077772743901137959722049, 4.17526339642412635099799215653, 4.50069135023256668013015384362, 5.39890127734916384538248571798, 6.29902244239417238411711653275, 6.77342476693903807043098489924, 7.10231972777536168992340381319, 7.52904831264959363218795114306, 8.102842478582627260086737971183, 8.382821401789467171793381100729, 8.976558111229072345918055801937, 9.491872697363616112831980572770, 9.973648641765421886781701531210, 10.44819065425930979156485156588, 11.21782759606127006318309868009, 11.27437239494547146492453416813

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