Properties

Label 4-910e2-1.1-c1e2-0-1
Degree $4$
Conductor $828100$
Sign $1$
Analytic cond. $52.8003$
Root an. cond. $2.69562$
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  − 2·2-s + 3·4-s − 5-s − 7-s − 4·8-s + 3·9-s + 2·10-s − 11-s − 5·13-s + 2·14-s + 5·16-s − 8·17-s − 6·18-s − 8·19-s − 3·20-s + 2·22-s + 10·23-s + 10·26-s − 3·28-s + 2·29-s − 10·31-s − 6·32-s + 16·34-s + 35-s + 9·36-s − 4·37-s + 16·38-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.447·5-s − 0.377·7-s − 1.41·8-s + 9-s + 0.632·10-s − 0.301·11-s − 1.38·13-s + 0.534·14-s + 5/4·16-s − 1.94·17-s − 1.41·18-s − 1.83·19-s − 0.670·20-s + 0.426·22-s + 2.08·23-s + 1.96·26-s − 0.566·28-s + 0.371·29-s − 1.79·31-s − 1.06·32-s + 2.74·34-s + 0.169·35-s + 3/2·36-s − 0.657·37-s + 2.59·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(828100\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(52.8003\)
Root analytic conductor: \(2.69562\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 828100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1726577376\)
\(L(\frac12)\) \(\approx\) \(0.1726577376\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + T + p T^{2} \)
13$C_2$ \( 1 + 5 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
11$C_2^2$ \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 5 T - 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 12 T + 73 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 16 T + 183 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 2 T - 75 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−10.52796594695979878199182042631, −9.795950206284353913610816065733, −9.459468581265115748865655139313, −8.991866193509581368405682127914, −8.795935821127456575658625617291, −8.134446316029763702820524619344, −8.043989383862278072754168478459, −7.11109157167891147442845683911, −7.05627752997572032139066521326, −6.61300089150476102490955354780, −6.60123053386520552157316841037, −5.32919773760748920058169549670, −5.22845520551690744136524376096, −4.37300743878545598567081083930, −4.14111075344903286009663480685, −3.24893740290304536718548029527, −2.67788104112472863392467872818, −2.07001723757237341404615016693, −1.57504599189052410109176622911, −0.23932874241198483053841117808, 0.23932874241198483053841117808, 1.57504599189052410109176622911, 2.07001723757237341404615016693, 2.67788104112472863392467872818, 3.24893740290304536718548029527, 4.14111075344903286009663480685, 4.37300743878545598567081083930, 5.22845520551690744136524376096, 5.32919773760748920058169549670, 6.60123053386520552157316841037, 6.61300089150476102490955354780, 7.05627752997572032139066521326, 7.11109157167891147442845683911, 8.043989383862278072754168478459, 8.134446316029763702820524619344, 8.795935821127456575658625617291, 8.991866193509581368405682127914, 9.459468581265115748865655139313, 9.795950206284353913610816065733, 10.52796594695979878199182042631

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