Properties

Label 4-3234e2-1.1-c1e2-0-3
Degree $4$
Conductor $10458756$
Sign $1$
Analytic cond. $666.859$
Root an. cond. $5.08169$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·3-s + 3·4-s + 2·5-s + 4·6-s + 4·8-s + 3·9-s + 4·10-s − 2·11-s + 6·12-s + 4·15-s + 5·16-s + 6·17-s + 6·18-s + 2·19-s + 6·20-s − 4·22-s + 4·23-s + 8·24-s − 2·25-s + 4·27-s + 8·30-s + 10·31-s + 6·32-s − 4·33-s + 12·34-s + 9·36-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s + 1.41·8-s + 9-s + 1.26·10-s − 0.603·11-s + 1.73·12-s + 1.03·15-s + 5/4·16-s + 1.45·17-s + 1.41·18-s + 0.458·19-s + 1.34·20-s − 0.852·22-s + 0.834·23-s + 1.63·24-s − 2/5·25-s + 0.769·27-s + 1.46·30-s + 1.79·31-s + 1.06·32-s − 0.696·33-s + 2.05·34-s + 3/2·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10458756\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(666.859\)
Root analytic conductor: \(5.08169\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3234} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10458756,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(17.35343833\)
\(L(\frac12)\) \(\approx\) \(17.35343833\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_4$ \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 10 T + 82 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_4$ \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 2 T + 162 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 4 T + 162 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + p T^{2} )^{2} \)
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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

−8.741788341757619207695356282185, −8.406468254377956189503924274561, −7.88558127539567505287852571959, −7.77917153838629961445706251782, −7.35301248615752742884404015787, −6.96589949112444028613204410035, −6.36407392004671373273687794092, −6.30087679671376595225245916209, −5.62760158844231651230699794767, −5.44482889569002572148455740427, −4.84100887330766401618618991845, −4.83359683909523356522985881653, −4.00835850179094160703005389683, −3.70116916744900971294743122738, −3.33255447285717329522932116873, −2.82355838547928644173038054560, −2.34315154550203202784495590440, −2.30622061104618036960300448251, −1.30339527837106869667181697221, −1.04108303344521762954040797548, 1.04108303344521762954040797548, 1.30339527837106869667181697221, 2.30622061104618036960300448251, 2.34315154550203202784495590440, 2.82355838547928644173038054560, 3.33255447285717329522932116873, 3.70116916744900971294743122738, 4.00835850179094160703005389683, 4.83359683909523356522985881653, 4.84100887330766401618618991845, 5.44482889569002572148455740427, 5.62760158844231651230699794767, 6.30087679671376595225245916209, 6.36407392004671373273687794092, 6.96589949112444028613204410035, 7.35301248615752742884404015787, 7.77917153838629961445706251782, 7.88558127539567505287852571959, 8.406468254377956189503924274561, 8.741788341757619207695356282185

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