Properties

Label 8-3360e4-1.1-c0e4-0-3
Degree $8$
Conductor $1.275\times 10^{14}$
Sign $1$
Analytic cond. $7.90652$
Root an. cond. $1.29493$
Motivic weight $0$
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  − 2·49-s + 4·73-s − 81-s + 4·97-s − 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  − 2·49-s + 4·73-s − 81-s + 4·97-s − 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(7.90652\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3360} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.367900537\)
\(L(\frac12)\) \(\approx\) \(1.367900537\)
\(L(1)\) 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 \( 1 \)
3$C_2^2$ \( 1 + T^{4} \)
5$C_2^2$ \( 1 + T^{4} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + T^{4} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.37422552127631811990744904537, −6.07003763132055325354077506711, −6.04811878382530264620116410581, −5.64060467304776256336655797492, −5.33667137630832365099942128043, −5.27409178840062880015855129755, −5.19414552198861979278733770581, −4.99406100000956186268089200398, −4.54888044872619501393177881343, −4.49038104802090756636012496738, −4.36764642988661584798371395177, −4.00508583512675145322602768476, −3.68317865678735530973936744716, −3.68271564241593316547300509276, −3.46433824453624529971639297433, −3.04580474335413517861486799073, −2.86572896875207406676855758742, −2.84176195310126380926635618530, −2.30594605071210871582426499340, −2.16157231065177360550707366089, −1.87219009593183543242796165403, −1.69352324724406189620353224899, −1.24973401361455021484305002856, −0.973027253471665838444414465238, −0.48976174427291262581174966561, 0.48976174427291262581174966561, 0.973027253471665838444414465238, 1.24973401361455021484305002856, 1.69352324724406189620353224899, 1.87219009593183543242796165403, 2.16157231065177360550707366089, 2.30594605071210871582426499340, 2.84176195310126380926635618530, 2.86572896875207406676855758742, 3.04580474335413517861486799073, 3.46433824453624529971639297433, 3.68271564241593316547300509276, 3.68317865678735530973936744716, 4.00508583512675145322602768476, 4.36764642988661584798371395177, 4.49038104802090756636012496738, 4.54888044872619501393177881343, 4.99406100000956186268089200398, 5.19414552198861979278733770581, 5.27409178840062880015855129755, 5.33667137630832365099942128043, 5.64060467304776256336655797492, 6.04811878382530264620116410581, 6.07003763132055325354077506711, 6.37422552127631811990744904537

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