Properties

Label 4-2156e2-1.1-c1e2-0-8
Degree $4$
Conductor $4648336$
Sign $1$
Analytic cond. $296.381$
Root an. cond. $4.14918$
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  − 4·9-s − 2·11-s − 8·23-s − 10·25-s + 4·29-s − 8·37-s + 4·43-s − 8·53-s − 16·67-s − 20·79-s + 7·81-s + 8·99-s − 24·107-s − 20·109-s + 28·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 24·169-s + 173-s + ⋯
L(s)  = 1  − 4/3·9-s − 0.603·11-s − 1.66·23-s − 2·25-s + 0.742·29-s − 1.31·37-s + 0.609·43-s − 1.09·53-s − 1.95·67-s − 2.25·79-s + 7/9·81-s + 0.804·99-s − 2.32·107-s − 1.91·109-s + 2.63·113-s + 3/11·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.84·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4648336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4648336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4648336\)    =    \(2^{4} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(296.381\)
Root analytic conductor: \(4.14918\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 4648336,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 144 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
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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

−8.853688637079875985672995107325, −8.440589346773426646628120495820, −8.079947665412074662667541496867, −7.88337627221818070900287019478, −7.39138448346768133803287054561, −7.03501481534130683314167794213, −6.33081261151516045690573580417, −6.09474967324740660128057137710, −5.67679028033406522090451404935, −5.54058176239885311907566078614, −4.80664797300500004375766506422, −4.51116596159638844738708667211, −3.76637679943203824314287076897, −3.66134323621632491736547069882, −2.78826958433702820106589383271, −2.64926589196412890908447145525, −1.93169039622124833986791974313, −1.41410101917632202608535330189, 0, 0, 1.41410101917632202608535330189, 1.93169039622124833986791974313, 2.64926589196412890908447145525, 2.78826958433702820106589383271, 3.66134323621632491736547069882, 3.76637679943203824314287076897, 4.51116596159638844738708667211, 4.80664797300500004375766506422, 5.54058176239885311907566078614, 5.67679028033406522090451404935, 6.09474967324740660128057137710, 6.33081261151516045690573580417, 7.03501481534130683314167794213, 7.39138448346768133803287054561, 7.88337627221818070900287019478, 8.079947665412074662667541496867, 8.440589346773426646628120495820, 8.853688637079875985672995107325

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