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Properties

Label 4-1224e2-1.1-c0e2-0-2
Degree $4$
Conductor $1498176$
Sign $1$
Analytic cond. $0.373144$
Root an. cond. $0.781572$
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  − 2·2-s + 3·4-s − 4·8-s + 5·16-s + 2·17-s − 6·32-s − 4·34-s − 4·43-s + 7·64-s + 6·68-s + 8·86-s + 4·89-s + 2·121-s + 127-s − 8·128-s + 131-s − 8·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s − 12·172-s + 173-s + ⋯
L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 5·16-s + 2·17-s − 6·32-s − 4·34-s − 4·43-s + 7·64-s + 6·68-s + 8·86-s + 4·89-s + 2·121-s + 127-s − 8·128-s + 131-s − 8·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s − 12·172-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1498176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1498176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1498176\)    =    \(2^{6} \cdot 3^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(0.373144\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1498176,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4396707309\)
\(L(\frac12)\) \(\approx\) \(0.4396707309\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
17$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$ \( ( 1 + T )^{4} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_1$ \( ( 1 - T )^{4} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−9.926047464720743482765543244786, −9.921096026502799667245284448088, −9.172632880674522076251535293166, −9.089320330054712159635694496889, −8.338234300555155476379446874638, −8.262372192001780137580380421882, −7.67742463348911590685979357817, −7.61618971070998471370021541179, −6.78644340089740919807353642420, −6.74610032239481169120774554765, −6.16568914155503284830000562469, −5.65405133874484537786649750719, −5.27787217812789025639869372087, −4.71575497748483378375653844011, −3.58716429165084320403398608547, −3.36709884565606494745634365404, −2.93934477872671429160988624196, −2.02353823104298509092755551481, −1.65444733736732027359488382683, −0.835860832016285951941877200020, 0.835860832016285951941877200020, 1.65444733736732027359488382683, 2.02353823104298509092755551481, 2.93934477872671429160988624196, 3.36709884565606494745634365404, 3.58716429165084320403398608547, 4.71575497748483378375653844011, 5.27787217812789025639869372087, 5.65405133874484537786649750719, 6.16568914155503284830000562469, 6.74610032239481169120774554765, 6.78644340089740919807353642420, 7.61618971070998471370021541179, 7.67742463348911590685979357817, 8.262372192001780137580380421882, 8.338234300555155476379446874638, 9.089320330054712159635694496889, 9.172632880674522076251535293166, 9.921096026502799667245284448088, 9.926047464720743482765543244786

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