Properties

Label 4-2790e2-1.1-c1e2-0-0
Degree $4$
Conductor $7784100$
Sign $1$
Analytic cond. $496.320$
Root an. cond. $4.71998$
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  − 4-s − 2·5-s + 2·11-s + 16-s − 6·19-s + 2·20-s − 25-s + 12·29-s − 2·31-s − 4·41-s − 2·44-s − 11·49-s − 4·55-s − 28·59-s + 28·61-s − 64-s − 18·71-s + 6·76-s − 30·79-s − 2·80-s − 2·89-s + 12·95-s + 100-s − 14·101-s − 20·109-s − 12·116-s − 19·121-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.894·5-s + 0.603·11-s + 1/4·16-s − 1.37·19-s + 0.447·20-s − 1/5·25-s + 2.22·29-s − 0.359·31-s − 0.624·41-s − 0.301·44-s − 1.57·49-s − 0.539·55-s − 3.64·59-s + 3.58·61-s − 1/8·64-s − 2.13·71-s + 0.688·76-s − 3.37·79-s − 0.223·80-s − 0.211·89-s + 1.23·95-s + 1/10·100-s − 1.39·101-s − 1.91·109-s − 1.11·116-s − 1.72·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7784100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7784100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7784100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(496.320\)
Root analytic conductor: \(4.71998\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7784100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3274984437\)
\(L(\frac12)\) \(\approx\) \(0.3274984437\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
31$C_1$ \( ( 1 + T )^{2} \)
good7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−8.886219413233937066456271545131, −8.530706566840508756791543044039, −8.271669679684392021725819810208, −8.013839022563884743813875916976, −7.58832181191334339369297202948, −6.92782289578653014125952919763, −6.87405329908167991431624096935, −6.32715308451551047053990173459, −6.02428768007561862407224937574, −5.53799505196582219327969592858, −4.86628851339447821948924205768, −4.70715263235650874966324045221, −4.21683845573726497770047438271, −3.91643077326064205635575686963, −3.51240918904382389745026195888, −2.72925955331772984190899610736, −2.68473719931766097413205363077, −1.53702074080884874746533536959, −1.34960767249870833986809211380, −0.19353611109792239249739387624, 0.19353611109792239249739387624, 1.34960767249870833986809211380, 1.53702074080884874746533536959, 2.68473719931766097413205363077, 2.72925955331772984190899610736, 3.51240918904382389745026195888, 3.91643077326064205635575686963, 4.21683845573726497770047438271, 4.70715263235650874966324045221, 4.86628851339447821948924205768, 5.53799505196582219327969592858, 6.02428768007561862407224937574, 6.32715308451551047053990173459, 6.87405329908167991431624096935, 6.92782289578653014125952919763, 7.58832181191334339369297202948, 8.013839022563884743813875916976, 8.271669679684392021725819810208, 8.530706566840508756791543044039, 8.886219413233937066456271545131

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