Properties

Label 4-836e2-1.1-c1e2-0-14
Degree $4$
Conductor $698896$
Sign $1$
Analytic cond. $44.5622$
Root an. cond. $2.58369$
Motivic weight $1$
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  + 4·3-s − 2·5-s + 6·9-s + 5·11-s − 8·15-s + 16·23-s − 7·25-s − 4·27-s + 8·31-s + 20·33-s + 20·37-s − 12·45-s − 2·47-s − 5·49-s − 8·53-s − 10·55-s + 12·59-s − 24·67-s + 64·69-s + 4·71-s − 28·75-s − 37·81-s + 24·89-s + 32·93-s − 16·97-s + 30·99-s − 12·103-s + ⋯
L(s)  = 1  + 2.30·3-s − 0.894·5-s + 2·9-s + 1.50·11-s − 2.06·15-s + 3.33·23-s − 7/5·25-s − 0.769·27-s + 1.43·31-s + 3.48·33-s + 3.28·37-s − 1.78·45-s − 0.291·47-s − 5/7·49-s − 1.09·53-s − 1.34·55-s + 1.56·59-s − 2.93·67-s + 7.70·69-s + 0.474·71-s − 3.23·75-s − 4.11·81-s + 2.54·89-s + 3.31·93-s − 1.62·97-s + 3.01·99-s − 1.18·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(698896\)    =    \(2^{4} \cdot 11^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(44.5622\)
Root analytic conductor: \(2.58369\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 698896,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.369608332\)
\(L(\frac12)\) \(\approx\) \(4.369608332\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_2$ \( 1 - 5 T + p T^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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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

−8.305060519445702337530152822868, −7.933665480997825519415597461102, −7.62429229152375892340957919668, −7.32424896802954806677465404753, −6.44587861158160350440717785391, −6.44323514092267885981017350010, −5.57209357785375076961912739494, −4.85244736948764917088509248105, −4.15537276955970084853193543262, −4.08026045073955845688679172688, −3.36880239235033917183403851835, −2.82511261070703297705797963037, −2.77013505033142815042798931153, −1.74404941065737830362193213595, −0.971870007004528213384011971816, 0.971870007004528213384011971816, 1.74404941065737830362193213595, 2.77013505033142815042798931153, 2.82511261070703297705797963037, 3.36880239235033917183403851835, 4.08026045073955845688679172688, 4.15537276955970084853193543262, 4.85244736948764917088509248105, 5.57209357785375076961912739494, 6.44323514092267885981017350010, 6.44587861158160350440717785391, 7.32424896802954806677465404753, 7.62429229152375892340957919668, 7.933665480997825519415597461102, 8.305060519445702337530152822868

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