Properties

Label 4-6e8-1.1-c1e2-0-35
Degree $4$
Conductor $1679616$
Sign $1$
Analytic cond. $107.093$
Root an. cond. $3.21692$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 2·13-s − 16·19-s + 5·25-s − 4·31-s − 20·37-s + 8·43-s + 7·49-s − 14·61-s − 16·67-s − 20·73-s − 4·79-s + 8·91-s − 14·97-s + 20·103-s + 4·109-s + 11·121-s + 127-s + 131-s + 64·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1.51·7-s − 0.554·13-s − 3.67·19-s + 25-s − 0.718·31-s − 3.28·37-s + 1.21·43-s + 49-s − 1.79·61-s − 1.95·67-s − 2.34·73-s − 0.450·79-s + 0.838·91-s − 1.42·97-s + 1.97·103-s + 0.383·109-s + 121-s + 0.0887·127-s + 0.0873·131-s + 5.54·133-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1679616\)    =    \(2^{8} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(107.093\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1296} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1679616,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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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

−9.211713298004280934576305971546, −8.990890859707656633183436792477, −8.802252600257081365423287630570, −8.483252289245808846100882759500, −7.73338015764669673333066923383, −7.33345691721823336896265067753, −6.86841691726000487303562102600, −6.63272656572301314153677296946, −6.03614083244652445333918166232, −6.02312489529303817182564551620, −5.17631427365747618133931549080, −4.75215650835674560577797473830, −4.06553442717947310899147176976, −4.00417543536580328788535595394, −3.06771075596467426674622927791, −2.89245730339382694666960367400, −2.09217505903120938420458680056, −1.61736177436991935904221204202, 0, 0, 1.61736177436991935904221204202, 2.09217505903120938420458680056, 2.89245730339382694666960367400, 3.06771075596467426674622927791, 4.00417543536580328788535595394, 4.06553442717947310899147176976, 4.75215650835674560577797473830, 5.17631427365747618133931549080, 6.02312489529303817182564551620, 6.03614083244652445333918166232, 6.63272656572301314153677296946, 6.86841691726000487303562102600, 7.33345691721823336896265067753, 7.73338015764669673333066923383, 8.483252289245808846100882759500, 8.802252600257081365423287630570, 8.990890859707656633183436792477, 9.211713298004280934576305971546

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