Properties

Label 4-608e2-1.1-c1e2-0-9
Degree $4$
Conductor $369664$
Sign $1$
Analytic cond. $23.5700$
Root an. cond. $2.20338$
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  + 4·3-s + 6·9-s + 6·11-s + 10·17-s + 2·19-s − 9·25-s − 4·27-s + 24·33-s + 12·41-s + 14·43-s − 5·49-s + 40·51-s + 8·57-s − 28·59-s − 30·73-s − 36·75-s − 37·81-s − 8·83-s + 32·97-s + 36·99-s − 20·107-s + 4·113-s + 5·121-s + 48·123-s + 127-s + 56·129-s + 131-s + ⋯
L(s)  = 1  + 2.30·3-s + 2·9-s + 1.80·11-s + 2.42·17-s + 0.458·19-s − 9/5·25-s − 0.769·27-s + 4.17·33-s + 1.87·41-s + 2.13·43-s − 5/7·49-s + 5.60·51-s + 1.05·57-s − 3.64·59-s − 3.51·73-s − 4.15·75-s − 4.11·81-s − 0.878·83-s + 3.24·97-s + 3.61·99-s − 1.93·107-s + 0.376·113-s + 5/11·121-s + 4.32·123-s + 0.0887·127-s + 4.93·129-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(369664\)    =    \(2^{10} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(23.5700\)
Root analytic conductor: \(2.20338\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{369664} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 369664,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.804397963\)
\(L(\frac12)\) \(\approx\) \(4.804397963\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
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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

−8.865784190712433894154140882392, −8.186562113914066026677579677805, −7.79236899027639545070642498110, −7.43537131861627444263317650332, −7.41110136347227768123475878009, −6.20879084669067007942487517486, −5.89904763388088518363904025264, −5.62405161804576690819108289841, −4.35808362404597816678481770906, −4.19084046620099313866498183608, −3.41145508666670040670114085640, −3.25598629803065603741482539939, −2.67622771685970545269516699451, −1.81226351589066908089801551987, −1.26674131115266326801058270565, 1.26674131115266326801058270565, 1.81226351589066908089801551987, 2.67622771685970545269516699451, 3.25598629803065603741482539939, 3.41145508666670040670114085640, 4.19084046620099313866498183608, 4.35808362404597816678481770906, 5.62405161804576690819108289841, 5.89904763388088518363904025264, 6.20879084669067007942487517486, 7.41110136347227768123475878009, 7.43537131861627444263317650332, 7.79236899027639545070642498110, 8.186562113914066026677579677805, 8.865784190712433894154140882392

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