Properties

Label 4-2842e2-1.1-c1e2-0-6
Degree $4$
Conductor $8076964$
Sign $1$
Analytic cond. $514.994$
Root an. cond. $4.76376$
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  + 2·2-s + 3·4-s + 4·8-s − 6·9-s − 4·11-s + 5·16-s − 12·18-s − 8·22-s − 12·23-s − 8·25-s − 2·29-s + 6·32-s − 18·36-s − 8·37-s − 8·43-s − 12·44-s − 24·46-s − 16·50-s − 12·53-s − 4·58-s + 7·64-s + 8·67-s − 4·71-s − 24·72-s − 16·74-s − 24·79-s + 27·81-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s − 2·9-s − 1.20·11-s + 5/4·16-s − 2.82·18-s − 1.70·22-s − 2.50·23-s − 8/5·25-s − 0.371·29-s + 1.06·32-s − 3·36-s − 1.31·37-s − 1.21·43-s − 1.80·44-s − 3.53·46-s − 2.26·50-s − 1.64·53-s − 0.525·58-s + 7/8·64-s + 0.977·67-s − 0.474·71-s − 2.82·72-s − 1.85·74-s − 2.70·79-s + 3·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8076964 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8076964 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8076964\)    =    \(2^{2} \cdot 7^{4} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(514.994\)
Root analytic conductor: \(4.76376\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 8076964,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
29$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 60 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 116 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 128 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 176 T^{2} + p^{2} T^{4} \)
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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

−8.391380315410183321566307985131, −8.134192960944930627699696299254, −7.75125707506951203302501292996, −7.69206300215806924993208924290, −6.77477756733577158503456457859, −6.68087583947254292067490665931, −5.98782726668821337417103714650, −5.85590396186397951401981424018, −5.44380380667879276791351701699, −5.34721497262056777570880583713, −4.69835287911109903932785863065, −4.26471953308040053037934590732, −3.71068413147365303744513046607, −3.48091181158175284480969721410, −2.84903477660818846975483656186, −2.62397009510143994678621541863, −1.94548932682159096552640825297, −1.70828781096723838430056037580, 0, 0, 1.70828781096723838430056037580, 1.94548932682159096552640825297, 2.62397009510143994678621541863, 2.84903477660818846975483656186, 3.48091181158175284480969721410, 3.71068413147365303744513046607, 4.26471953308040053037934590732, 4.69835287911109903932785863065, 5.34721497262056777570880583713, 5.44380380667879276791351701699, 5.85590396186397951401981424018, 5.98782726668821337417103714650, 6.68087583947254292067490665931, 6.77477756733577158503456457859, 7.69206300215806924993208924290, 7.75125707506951203302501292996, 8.134192960944930627699696299254, 8.391380315410183321566307985131

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