Properties

Label 4-1078e2-1.1-c1e2-0-0
Degree $4$
Conductor $1162084$
Sign $1$
Analytic cond. $74.0954$
Root an. cond. $2.93391$
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  + 2-s − 4·5-s − 8-s + 3·9-s − 4·10-s + 11-s − 4·13-s − 16-s − 4·17-s + 3·18-s − 6·19-s + 22-s − 4·23-s + 5·25-s − 4·26-s − 4·29-s − 2·31-s − 4·34-s − 10·37-s − 6·38-s + 4·40-s − 8·41-s − 16·43-s − 12·45-s − 4·46-s + 2·47-s + 5·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.78·5-s − 0.353·8-s + 9-s − 1.26·10-s + 0.301·11-s − 1.10·13-s − 1/4·16-s − 0.970·17-s + 0.707·18-s − 1.37·19-s + 0.213·22-s − 0.834·23-s + 25-s − 0.784·26-s − 0.742·29-s − 0.359·31-s − 0.685·34-s − 1.64·37-s − 0.973·38-s + 0.632·40-s − 1.24·41-s − 2.43·43-s − 1.78·45-s − 0.589·46-s + 0.291·47-s + 0.707·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1162084\)    =    \(2^{2} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(74.0954\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1078} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1162084,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1503238438\)
\(L(\frac12)\) \(\approx\) \(0.1503238438\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
11$C_2$ \( 1 - T + T^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−10.08000033787343492297607970989, −9.875038947519482512342610663379, −9.133009965674375628519104596176, −8.675465696989419823372271681782, −8.559011360012276137600174435808, −7.905878418643774710519691748612, −7.48860344856437729135364096388, −7.22680704622501398107131237885, −6.85635868816809327932155916651, −6.11518121962573745494109512909, −6.11054143141054975237765970775, −4.86054059875307597403338837200, −4.69961615649638562593921852332, −4.68253937992556196726861317861, −3.73793485382853907577824326267, −3.66503930610902484547606492912, −3.18515153612322293879083860020, −1.92952037325567972179124686277, −1.92525314806828388113684110865, −0.14636214555811217611461498834, 0.14636214555811217611461498834, 1.92525314806828388113684110865, 1.92952037325567972179124686277, 3.18515153612322293879083860020, 3.66503930610902484547606492912, 3.73793485382853907577824326267, 4.68253937992556196726861317861, 4.69961615649638562593921852332, 4.86054059875307597403338837200, 6.11054143141054975237765970775, 6.11518121962573745494109512909, 6.85635868816809327932155916651, 7.22680704622501398107131237885, 7.48860344856437729135364096388, 7.905878418643774710519691748612, 8.559011360012276137600174435808, 8.675465696989419823372271681782, 9.133009965674375628519104596176, 9.875038947519482512342610663379, 10.08000033787343492297607970989

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