Properties

Label 4-133e2-1.1-c0e2-0-0
Degree $4$
Conductor $17689$
Sign $1$
Analytic cond. $0.00440572$
Root an. cond. $0.257634$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 5-s − 7-s − 9-s + 11-s − 2·17-s − 19-s − 20-s + 23-s + 25-s + 28-s − 35-s + 36-s − 2·43-s − 44-s − 45-s + 47-s + 55-s + 61-s + 63-s + 64-s + 2·68-s + 73-s + 76-s − 77-s − 2·83-s − 2·85-s + ⋯
L(s)  = 1  − 4-s + 5-s − 7-s − 9-s + 11-s − 2·17-s − 19-s − 20-s + 23-s + 25-s + 28-s − 35-s + 36-s − 2·43-s − 44-s − 45-s + 47-s + 55-s + 61-s + 63-s + 64-s + 2·68-s + 73-s + 76-s − 77-s − 2·83-s − 2·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(17689\)    =    \(7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.00440572\)
Root analytic conductor: \(0.257634\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{133} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 17689,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3307553836\)
\(L(\frac12)\) \(\approx\) \(0.3307553836\)
\(L(1)\) 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)$
bad7$C_2$ \( 1 + T + T^{2} \)
19$C_2$ \( 1 + T + T^{2} \)
good2$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
3$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
5$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2$ \( ( 1 + T + T^{2} )^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 + T + T^{2} )^{2} \)
47$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
79$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_2$ \( ( 1 + T + T^{2} )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−13.51536939115785482243110465220, −13.42880854240873474245589123863, −12.82004215862498781437345667003, −12.61363320452011912066508697198, −11.46539040593524054685473501032, −11.42606310251343268145856726421, −10.59528197589011550738162946023, −10.10117837795661673178713172310, −9.311234802124883832962565317195, −9.239794672657171368592271462543, −8.586299636721084614348707081449, −8.448405537237828755980821676130, −6.94213003987707110179046690510, −6.69334998687290640538404127101, −6.18638201546974727841309198593, −5.42331299502316119487515539401, −4.70809796419164559349036462342, −4.06434645796077638680479019311, −3.09997349348192763214703328180, −2.16502060743839806249047133845, 2.16502060743839806249047133845, 3.09997349348192763214703328180, 4.06434645796077638680479019311, 4.70809796419164559349036462342, 5.42331299502316119487515539401, 6.18638201546974727841309198593, 6.69334998687290640538404127101, 6.94213003987707110179046690510, 8.448405537237828755980821676130, 8.586299636721084614348707081449, 9.239794672657171368592271462543, 9.311234802124883832962565317195, 10.10117837795661673178713172310, 10.59528197589011550738162946023, 11.42606310251343268145856726421, 11.46539040593524054685473501032, 12.61363320452011912066508697198, 12.82004215862498781437345667003, 13.42880854240873474245589123863, 13.51536939115785482243110465220

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