Properties

Label 4-2829888-1.1-c1e2-0-7
Degree $4$
Conductor $2829888$
Sign $1$
Analytic cond. $180.436$
Root an. cond. $3.66505$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·3-s − 4-s − 2·6-s + 3·8-s + 3·9-s + 8·11-s − 2·12-s − 16-s + 17-s − 3·18-s − 8·19-s − 8·22-s + 6·24-s + 2·25-s + 4·27-s − 5·32-s + 16·33-s − 34-s − 3·36-s + 8·38-s + 8·43-s − 8·44-s − 2·48-s + 2·49-s − 2·50-s + 2·51-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.816·6-s + 1.06·8-s + 9-s + 2.41·11-s − 0.577·12-s − 1/4·16-s + 0.242·17-s − 0.707·18-s − 1.83·19-s − 1.70·22-s + 1.22·24-s + 2/5·25-s + 0.769·27-s − 0.883·32-s + 2.78·33-s − 0.171·34-s − 1/2·36-s + 1.29·38-s + 1.21·43-s − 1.20·44-s − 0.288·48-s + 2/7·49-s − 0.282·50-s + 0.280·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2829888 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2829888 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2829888\)    =    \(2^{6} \cdot 3^{2} \cdot 17^{3}\)
Sign: $1$
Analytic conductor: \(180.436\)
Root analytic conductor: \(3.66505\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2829888} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2829888,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.735936085\)
\(L(\frac12)\) \(\approx\) \(2.735936085\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
17$C_1$ \( 1 - T \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
61$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
71$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−7.82580289195797930220355288431, −7.14257618232615439309453339322, −6.81089493407269657838307778384, −6.66687350936944381817930751537, −5.97184583023480145443864217661, −5.59334879276053026033607263499, −4.73157702271605766153507759357, −4.38066853020012071837140433024, −4.14871936049865192617863162664, −3.58805876328343905426425196162, −3.33506544840499141212096962726, −2.28251357059639089877751737038, −2.04655503075998176246419168666, −1.28950375172625955369776789729, −0.75526794503091583672922454443, 0.75526794503091583672922454443, 1.28950375172625955369776789729, 2.04655503075998176246419168666, 2.28251357059639089877751737038, 3.33506544840499141212096962726, 3.58805876328343905426425196162, 4.14871936049865192617863162664, 4.38066853020012071837140433024, 4.73157702271605766153507759357, 5.59334879276053026033607263499, 5.97184583023480145443864217661, 6.66687350936944381817930751537, 6.81089493407269657838307778384, 7.14257618232615439309453339322, 7.82580289195797930220355288431

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