Properties

Label 4-1152e2-1.1-c1e2-0-49
Degree $4$
Conductor $1327104$
Sign $-1$
Analytic cond. $84.6173$
Root an. cond. $3.03294$
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  + 3-s − 2·9-s − 3·11-s + 6·19-s + 25-s − 5·27-s − 3·33-s − 3·41-s − 9·43-s + 49-s + 6·57-s − 3·59-s + 3·67-s + 2·73-s + 75-s + 81-s + 18·83-s − 11·97-s + 6·99-s + 12·107-s − 13·121-s − 3·123-s + 127-s − 9·129-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.577·3-s − 2/3·9-s − 0.904·11-s + 1.37·19-s + 1/5·25-s − 0.962·27-s − 0.522·33-s − 0.468·41-s − 1.37·43-s + 1/7·49-s + 0.794·57-s − 0.390·59-s + 0.366·67-s + 0.234·73-s + 0.115·75-s + 1/9·81-s + 1.97·83-s − 1.11·97-s + 0.603·99-s + 1.16·107-s − 1.18·121-s − 0.270·123-s + 0.0887·127-s − 0.792·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1327104\)    =    \(2^{14} \cdot 3^{4}\)
Sign: $-1$
Analytic conductor: \(84.6173\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1327104,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - T + p T^{2} \)
good5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \)
23$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
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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

−7.77414039684663719350713260941, −7.50985772346961920043737082893, −6.94603029229376880059626002517, −6.46956425642362934550354692684, −5.99631814471086446128048122260, −5.43833660309139788597201101972, −5.11164032886282461105931847026, −4.78931725740895564594288173538, −3.94660333775964115762263240533, −3.45588424706043277872090198087, −3.08147156795233720564054492608, −2.54476562420136383594593424615, −1.98551657033926118969802773471, −1.11555042852865242217690916956, 0, 1.11555042852865242217690916956, 1.98551657033926118969802773471, 2.54476562420136383594593424615, 3.08147156795233720564054492608, 3.45588424706043277872090198087, 3.94660333775964115762263240533, 4.78931725740895564594288173538, 5.11164032886282461105931847026, 5.43833660309139788597201101972, 5.99631814471086446128048122260, 6.46956425642362934550354692684, 6.94603029229376880059626002517, 7.50985772346961920043737082893, 7.77414039684663719350713260941

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