Properties

Label 4-642816-1.1-c1e2-0-4
Degree $4$
Conductor $642816$
Sign $-1$
Analytic cond. $40.9865$
Root an. cond. $2.53023$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 2·13-s − 10·19-s + 8·25-s + 3·31-s + 16·37-s + 10·43-s + 2·49-s − 4·61-s + 2·67-s + 10·73-s − 22·79-s + 8·91-s − 12·97-s + 6·103-s − 8·109-s + 6·121-s + 127-s + 131-s + 40·133-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 − 2.29·19-s + 8/5·25-s + 0.538·31-s + 2.63·37-s + 1.52·43-s + 2/7·49-s − 0.512·61-s + 0.244·67-s + 1.17·73-s − 2.47·79-s + 0.838·91-s − 1.21·97-s + 0.591·103-s − 0.766·109-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 3.46·133-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 642816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 642816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(642816\)    =    \(2^{8} \cdot 3^{4} \cdot 31\)
Sign: $-1$
Analytic conductor: \(40.9865\)
Root analytic conductor: \(2.53023\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{642816} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 642816,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
31$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
good5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
71$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−8.338246729726973369186549566247, −7.56808603094627896763977059923, −7.27843699959541616164857451731, −6.59752258775712761540915123316, −6.35535212975509318700393389423, −6.12944618666093432791197002319, −5.44760829983544462949766632329, −4.75799827232461868811658612184, −4.27488602160160640068851970824, −3.99217758383044073314889782404, −3.05982993545672452944983270049, −2.72409555190917276757649420611, −2.23322800034438911239426459783, −1.02697405316305980359653848060, 0, 1.02697405316305980359653848060, 2.23322800034438911239426459783, 2.72409555190917276757649420611, 3.05982993545672452944983270049, 3.99217758383044073314889782404, 4.27488602160160640068851970824, 4.75799827232461868811658612184, 5.44760829983544462949766632329, 6.12944618666093432791197002319, 6.35535212975509318700393389423, 6.59752258775712761540915123316, 7.27843699959541616164857451731, 7.56808603094627896763977059923, 8.338246729726973369186549566247

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