Properties

Label 4-1400e2-1.1-c0e2-0-5
Degree $4$
Conductor $1960000$
Sign $1$
Analytic cond. $0.488169$
Root an. cond. $0.835877$
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  − 2-s − 7-s + 8-s + 9-s + 14-s − 16-s + 3·17-s − 18-s − 23-s + 3·31-s − 3·34-s + 46-s + 3·47-s − 56-s − 3·62-s − 63-s + 64-s + 2·71-s + 72-s − 79-s + 3·89-s − 3·94-s − 3·103-s + 112-s − 2·113-s − 3·119-s − 121-s + ⋯
L(s)  = 1  − 2-s − 7-s + 8-s + 9-s + 14-s − 16-s + 3·17-s − 18-s − 23-s + 3·31-s − 3·34-s + 46-s + 3·47-s − 56-s − 3·62-s − 63-s + 64-s + 2·71-s + 72-s − 79-s + 3·89-s − 3·94-s − 3·103-s + 112-s − 2·113-s − 3·119-s − 121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1960000\)    =    \(2^{6} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.488169\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1960000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6839190028\)
\(L(\frac12)\) \(\approx\) \(0.6839190028\)
\(L(1)\) 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)$
bad2$C_2$ \( 1 + T + T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 + T + T^{2} \)
good3$C_2^2$ \( 1 - T^{2} + T^{4} \)
11$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
19$C_2^2$ \( 1 - T^{2} + T^{4} \)
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_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
37$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
41$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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$C_2^2$ \( 1 - T^{2} + T^{4} \)
61$C_2^2$ \( 1 - T^{2} + T^{4} \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_2$ \( ( 1 - T + T^{2} )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
97$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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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

−10.03356353414453699103718349205, −9.644215447676611310771336950373, −9.228480689166928959682455663129, −8.916920055650879232363472994235, −8.156488497678090914975982233093, −8.051645803199376206093487782013, −7.53180114653832798300357303879, −7.50301211291397246961275576037, −6.57608681935525486481620078292, −6.54981025084636754142039533373, −5.94927884281690563030827749250, −5.27167970366015021865604844389, −5.15349239292167639414577953861, −4.23784975096476487722017851375, −4.00983701707170394747777622196, −3.54415359655160071793357103862, −2.80887601638612593341476237134, −2.36846604699452952415481155773, −1.14555718080571493573882026365, −1.11729674022285314441964022612, 1.11729674022285314441964022612, 1.14555718080571493573882026365, 2.36846604699452952415481155773, 2.80887601638612593341476237134, 3.54415359655160071793357103862, 4.00983701707170394747777622196, 4.23784975096476487722017851375, 5.15349239292167639414577953861, 5.27167970366015021865604844389, 5.94927884281690563030827749250, 6.54981025084636754142039533373, 6.57608681935525486481620078292, 7.50301211291397246961275576037, 7.53180114653832798300357303879, 8.051645803199376206093487782013, 8.156488497678090914975982233093, 8.916920055650879232363472994235, 9.228480689166928959682455663129, 9.644215447676611310771336950373, 10.03356353414453699103718349205

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