Properties

Label 4-405e2-1.1-c1e2-0-13
Degree $4$
Conductor $164025$
Sign $1$
Analytic cond. $10.4583$
Root an. cond. $1.79831$
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  − 2·2-s + 2·4-s − 5-s − 4·8-s + 2·10-s − 5·11-s − 4·13-s + 8·16-s − 8·17-s − 10·19-s − 2·20-s + 10·22-s − 6·23-s + 8·26-s + 5·29-s + 9·31-s − 8·32-s + 16·34-s − 20·37-s + 20·38-s + 4·40-s − 7·41-s + 2·43-s − 10·44-s + 12·46-s − 2·47-s + 7·49-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.447·5-s − 1.41·8-s + 0.632·10-s − 1.50·11-s − 1.10·13-s + 2·16-s − 1.94·17-s − 2.29·19-s − 0.447·20-s + 2.13·22-s − 1.25·23-s + 1.56·26-s + 0.928·29-s + 1.61·31-s − 1.41·32-s + 2.74·34-s − 3.28·37-s + 3.24·38-s + 0.632·40-s − 1.09·41-s + 0.304·43-s − 1.50·44-s + 1.76·46-s − 0.291·47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(164025\)    =    \(3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(10.4583\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 164025,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad3 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
good2$C_2^2$ \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 7 T + 8 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - T - 58 T^{2} - p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 6 T - 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
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

−10.62294358370808802227290137830, −10.52755959451468544356437132672, −10.02796411348234986452558275639, −9.954365582767116809296842721188, −8.840443085737637672864224671104, −8.761550421534312872876249621153, −8.282841994599585323249472568018, −8.265369837306651636680986082671, −7.29090938036221534532074082949, −6.92315502455055926209526901200, −6.57561704634863464608148049758, −5.82378390722174223213057106595, −5.34129194771495489156303064358, −4.44088180038416228567280094359, −4.23024492524889618882193378555, −3.05208159602070366829036127000, −2.47450709069665290588655485953, −1.98958486613321615527244991942, 0, 0, 1.98958486613321615527244991942, 2.47450709069665290588655485953, 3.05208159602070366829036127000, 4.23024492524889618882193378555, 4.44088180038416228567280094359, 5.34129194771495489156303064358, 5.82378390722174223213057106595, 6.57561704634863464608148049758, 6.92315502455055926209526901200, 7.29090938036221534532074082949, 8.265369837306651636680986082671, 8.282841994599585323249472568018, 8.761550421534312872876249621153, 8.840443085737637672864224671104, 9.954365582767116809296842721188, 10.02796411348234986452558275639, 10.52755959451468544356437132672, 10.62294358370808802227290137830

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