Properties

Label 4-1078e2-1.1-c1e2-0-32
Degree $4$
Conductor $1162084$
Sign $-1$
Analytic cond. $74.0954$
Root an. cond. $2.93391$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 2·11-s + 16-s − 4·23-s − 4·25-s + 4·29-s + 4·37-s − 12·43-s − 2·44-s + 12·53-s + 64-s − 8·67-s + 4·71-s − 8·79-s − 9·81-s − 4·92-s − 4·100-s + 4·107-s + 8·109-s + 4·113-s + 4·116-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 4·148-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.603·11-s + 1/4·16-s − 0.834·23-s − 4/5·25-s + 0.742·29-s + 0.657·37-s − 1.82·43-s − 0.301·44-s + 1.64·53-s + 1/8·64-s − 0.977·67-s + 0.474·71-s − 0.900·79-s − 81-s − 0.417·92-s − 2/5·100-s + 0.386·107-s + 0.766·109-s + 0.376·113-s + 0.371·116-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.328·148-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1162084,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
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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

−7.82585823189187357170604761317, −7.37267416160518445083630758435, −7.03578761223120241468223372121, −6.46893383218739650714879998906, −6.05555950723124558549416641980, −5.72163738566068629322183583957, −5.14689268447569153357109092356, −4.70382268243491524243564776458, −4.11919614030930717296131916713, −3.63009233636465865592522959380, −3.02802781634539866723621945035, −2.46955742152867532593087765372, −1.96107296542132559442757402741, −1.17566718002933858315702025437, 0, 1.17566718002933858315702025437, 1.96107296542132559442757402741, 2.46955742152867532593087765372, 3.02802781634539866723621945035, 3.63009233636465865592522959380, 4.11919614030930717296131916713, 4.70382268243491524243564776458, 5.14689268447569153357109092356, 5.72163738566068629322183583957, 6.05555950723124558549416641980, 6.46893383218739650714879998906, 7.03578761223120241468223372121, 7.37267416160518445083630758435, 7.82585823189187357170604761317

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