Properties

Label 4-1078e2-1.1-c1e2-0-1
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s − 2·5-s − 2·6-s − 8-s + 3·9-s − 2·10-s − 11-s − 8·13-s + 4·15-s − 16-s + 3·18-s − 4·19-s − 22-s − 4·23-s + 2·24-s + 5·25-s − 8·26-s − 10·27-s + 4·29-s + 4·30-s + 10·31-s + 2·33-s + 6·37-s − 4·38-s + 16·39-s + 2·40-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 0.894·5-s − 0.816·6-s − 0.353·8-s + 9-s − 0.632·10-s − 0.301·11-s − 2.21·13-s + 1.03·15-s − 1/4·16-s + 0.707·18-s − 0.917·19-s − 0.213·22-s − 0.834·23-s + 0.408·24-s + 25-s − 1.56·26-s − 1.92·27-s + 0.742·29-s + 0.730·30-s + 1.79·31-s + 0.348·33-s + 0.986·37-s − 0.648·38-s + 2.56·39-s + 0.316·40-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

\(L(1)\) \(\approx\) \(0.2126230941\)
\(L(\frac12)\) \(\approx\) \(0.2126230941\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 10 T + 53 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 4 T - 57 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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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

−10.24803667058770397795027370933, −9.919402579247381229650396525367, −9.393050801438464190609356132860, −8.848230347662015276069176611433, −8.140560210250584206848825010845, −8.114974726979999944136498737233, −7.37428265443749302439275153552, −7.21502769034506298934328176616, −6.61994063891904614593524159116, −6.22260749729063194198420768014, −5.84619022743674287554604338135, −5.16844740100516949021650657759, −4.80258119780136087905492633840, −4.62101221991804696586040385955, −4.10491695588182596053675766811, −3.61307303972398834293074101973, −2.62900969894857450786351264314, −2.54773971534484745377811429033, −1.40038414670975303335191805557, −0.20032619055164613425568765078, 0.20032619055164613425568765078, 1.40038414670975303335191805557, 2.54773971534484745377811429033, 2.62900969894857450786351264314, 3.61307303972398834293074101973, 4.10491695588182596053675766811, 4.62101221991804696586040385955, 4.80258119780136087905492633840, 5.16844740100516949021650657759, 5.84619022743674287554604338135, 6.22260749729063194198420768014, 6.61994063891904614593524159116, 7.21502769034506298934328176616, 7.37428265443749302439275153552, 8.114974726979999944136498737233, 8.140560210250584206848825010845, 8.848230347662015276069176611433, 9.393050801438464190609356132860, 9.919402579247381229650396525367, 10.24803667058770397795027370933

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