Properties

Label 4-45e4-1.1-c1e2-0-5
Degree $4$
Conductor $4100625$
Sign $1$
Analytic cond. $261.459$
Root an. cond. $4.02115$
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  + 3·4-s + 4·11-s + 5·16-s + 16·19-s − 2·29-s − 10·41-s + 12·44-s + 5·49-s − 28·59-s + 14·61-s + 3·64-s − 4·71-s + 48·76-s + 12·79-s − 30·89-s + 36·101-s − 10·109-s − 6·116-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 3/2·4-s + 1.20·11-s + 5/4·16-s + 3.67·19-s − 0.371·29-s − 1.56·41-s + 1.80·44-s + 5/7·49-s − 3.64·59-s + 1.79·61-s + 3/8·64-s − 0.474·71-s + 5.50·76-s + 1.35·79-s − 3.17·89-s + 3.58·101-s − 0.957·109-s − 0.557·116-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4100625\)    =    \(3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(261.459\)
Root analytic conductor: \(4.02115\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4100625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.958741494\)
\(L(\frac12)\) \(\approx\) \(4.958741494\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 125 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−9.376741619708044686081009119557, −9.163013812254466075787784776182, −8.504759777416897739586786736708, −7.943172169087852326465739355537, −7.80973990263537273930878371406, −7.15597677106333918247873720315, −7.02473703030165987105606274057, −6.85542743206508358566982511708, −6.14943180023679515630897611102, −5.69687811853695285147992883523, −5.65579691115611211232008148623, −4.87293775156618201827285958311, −4.61755931869771490478403951750, −3.68037413688321536704585762883, −3.46407843211430183907768555112, −3.05726363888766807250643789256, −2.60315331284799429850266084139, −1.72838282775858378684817411189, −1.48514295959288926795682074441, −0.835991572129583043879403521465, 0.835991572129583043879403521465, 1.48514295959288926795682074441, 1.72838282775858378684817411189, 2.60315331284799429850266084139, 3.05726363888766807250643789256, 3.46407843211430183907768555112, 3.68037413688321536704585762883, 4.61755931869771490478403951750, 4.87293775156618201827285958311, 5.65579691115611211232008148623, 5.69687811853695285147992883523, 6.14943180023679515630897611102, 6.85542743206508358566982511708, 7.02473703030165987105606274057, 7.15597677106333918247873720315, 7.80973990263537273930878371406, 7.943172169087852326465739355537, 8.504759777416897739586786736708, 9.163013812254466075787784776182, 9.376741619708044686081009119557

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