Properties

Label 4-3920e2-1.1-c0e2-0-0
Degree $4$
Conductor $15366400$
Sign $1$
Analytic cond. $3.82724$
Root an. cond. $1.39869$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 9-s − 11-s − 2·13-s + 15-s + 17-s − 2·27-s − 2·29-s + 33-s + 2·39-s − 45-s − 47-s − 51-s + 55-s + 2·65-s − 4·71-s − 2·73-s − 79-s + 2·81-s − 4·83-s − 85-s + 2·87-s − 2·97-s − 99-s − 103-s + 109-s + ⋯
L(s)  = 1  − 3-s − 5-s + 9-s − 11-s − 2·13-s + 15-s + 17-s − 2·27-s − 2·29-s + 33-s + 2·39-s − 45-s − 47-s − 51-s + 55-s + 2·65-s − 4·71-s − 2·73-s − 79-s + 2·81-s − 4·83-s − 85-s + 2·87-s − 2·97-s − 99-s − 103-s + 109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15366400\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(3.82724\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3920} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15366400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001634087243\)
\(L(\frac12)\) \(\approx\) \(0.001634087243\)
\(L(1)\) 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 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
7 \( 1 \)
good3$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
11$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
13$C_2$ \( ( 1 + T + T^{2} )^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
29$C_2$ \( ( 1 + T + T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_1$ \( ( 1 + T )^{4} \)
73$C_2$ \( ( 1 + T + T^{2} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
83$C_1$ \( ( 1 + T )^{4} \)
89$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
97$C_2$ \( ( 1 + T + T^{2} )^{2} \)
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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

−9.208004666237471646548035870310, −8.372765247895775922316789846984, −7.86519428439525497819822141311, −7.53005861239001141794397994772, −7.40267394829386834699338797138, −7.24501194664841360452335092537, −6.80739465675325071096302125763, −6.06115396146323053010651760243, −5.70177951799821100173811433407, −5.54640364137133478784855125364, −5.21017622692175336726612236322, −4.60341898679440797292524763379, −4.35521299984307247806561806836, −4.04345882523985575165827446863, −3.44833510552664789025519452950, −2.87194927327832905627592696237, −2.65682932067581481402282493049, −1.69810432532893621843676467960, −1.51996097849080123173746990361, −0.02366696605190162706817059499, 0.02366696605190162706817059499, 1.51996097849080123173746990361, 1.69810432532893621843676467960, 2.65682932067581481402282493049, 2.87194927327832905627592696237, 3.44833510552664789025519452950, 4.04345882523985575165827446863, 4.35521299984307247806561806836, 4.60341898679440797292524763379, 5.21017622692175336726612236322, 5.54640364137133478784855125364, 5.70177951799821100173811433407, 6.06115396146323053010651760243, 6.80739465675325071096302125763, 7.24501194664841360452335092537, 7.40267394829386834699338797138, 7.53005861239001141794397994772, 7.86519428439525497819822141311, 8.372765247895775922316789846984, 9.208004666237471646548035870310

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