Properties

Label 4-5376e2-1.1-c1e2-0-42
Degree $4$
Conductor $28901376$
Sign $1$
Analytic cond. $1842.77$
Root an. cond. $6.55191$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 9-s − 4·17-s − 8·23-s + 6·25-s − 16·31-s + 4·41-s + 3·49-s + 2·63-s + 24·71-s − 4·73-s + 81-s − 12·89-s + 4·97-s − 16·103-s − 28·113-s + 8·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + 16·161-s + ⋯
L(s)  = 1  − 0.755·7-s − 1/3·9-s − 0.970·17-s − 1.66·23-s + 6/5·25-s − 2.87·31-s + 0.624·41-s + 3/7·49-s + 0.251·63-s + 2.84·71-s − 0.468·73-s + 1/9·81-s − 1.27·89-s + 0.406·97-s − 1.57·103-s − 2.63·113-s + 0.733·119-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.323·153-s + 0.0798·157-s + 1.26·161-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(28901376\)    =    \(2^{16} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1842.77\)
Root analytic conductor: \(6.55191\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 28901376,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−7.980429295829972763339150929376, −7.72213915376762324295688574673, −7.12848448210342444478720069643, −7.00005604742938908064987684111, −6.55197057222954709846150037150, −6.26426247803484509552425333969, −5.83618228497125908807436319511, −5.54797915635949854005537895631, −4.96385823607857016413063572385, −4.95201737198047055716805399331, −4.01473040705127168586474644964, −3.88383371842442626249107445350, −3.71516227340279091913054824687, −2.98805194554284973854058449427, −2.54705289132506810292487891514, −2.25480581246777942234856344863, −1.67371036811037920712114210785, −1.07402376152324713741471343362, 0, 0, 1.07402376152324713741471343362, 1.67371036811037920712114210785, 2.25480581246777942234856344863, 2.54705289132506810292487891514, 2.98805194554284973854058449427, 3.71516227340279091913054824687, 3.88383371842442626249107445350, 4.01473040705127168586474644964, 4.95201737198047055716805399331, 4.96385823607857016413063572385, 5.54797915635949854005537895631, 5.83618228497125908807436319511, 6.26426247803484509552425333969, 6.55197057222954709846150037150, 7.00005604742938908064987684111, 7.12848448210342444478720069643, 7.72213915376762324295688574673, 7.980429295829972763339150929376

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