Properties

Label 8-48e8-1.1-c2e4-0-8
Degree $8$
Conductor $2.818\times 10^{13}$
Sign $1$
Analytic cond. $1.55335\times 10^{7}$
Root an. cond. $7.92334$
Motivic weight $2$
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  − 16·13-s − 44·25-s − 112·37-s + 84·49-s − 304·61-s − 104·73-s + 72·97-s − 592·109-s + 420·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 516·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 1.23·13-s − 1.75·25-s − 3.02·37-s + 12/7·49-s − 4.98·61-s − 1.42·73-s + 0.742·97-s − 5.43·109-s + 3.47·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 3.05·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.55335\times 10^{7}\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2304} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 3^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1176508403\)
\(L(\frac12)\) \(\approx\) \(0.1176508403\)
\(L(2)\) 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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( ( 1 + 22 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 6 p T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 210 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p^{2} T^{2} )^{4} \)
17$C_2^2$ \( ( 1 + 130 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 174 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 546 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 1654 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 1418 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 28 T + p^{2} T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 670 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 2802 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 190 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 3350 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 3406 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 76 T + p^{2} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 5394 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 1890 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 26 T + p^{2} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 3702 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 334 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 14050 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p^{2} T^{2} )^{4} \)
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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

−6.18102686710989030614557826799, −6.15467368972563569625048305076, −5.82908937713698727993330295431, −5.47461825129319059647224827109, −5.40779196146490128799794546042, −5.14450316316397470035323669332, −5.06822274678705904002383288159, −4.56352036579030033409239311682, −4.56188431075377734340922051989, −4.33943079033002599979898731955, −4.25045981991544227666341731161, −3.57795477640566694159895759024, −3.56486009815332256011269847509, −3.48952358274270554625720193615, −3.33114278151335784027219914625, −2.72274782627156850661667108940, −2.55798766956239175810603131567, −2.34058607112579379910572704391, −2.31952847104946078522629684214, −1.68434288894091451819616701604, −1.49855655119243905425081630151, −1.38689434686113213569854057134, −1.03657755234955879368478042388, −0.22558777061047336091800918224, −0.094727468101646767921113531425, 0.094727468101646767921113531425, 0.22558777061047336091800918224, 1.03657755234955879368478042388, 1.38689434686113213569854057134, 1.49855655119243905425081630151, 1.68434288894091451819616701604, 2.31952847104946078522629684214, 2.34058607112579379910572704391, 2.55798766956239175810603131567, 2.72274782627156850661667108940, 3.33114278151335784027219914625, 3.48952358274270554625720193615, 3.56486009815332256011269847509, 3.57795477640566694159895759024, 4.25045981991544227666341731161, 4.33943079033002599979898731955, 4.56188431075377734340922051989, 4.56352036579030033409239311682, 5.06822274678705904002383288159, 5.14450316316397470035323669332, 5.40779196146490128799794546042, 5.47461825129319059647224827109, 5.82908937713698727993330295431, 6.15467368972563569625048305076, 6.18102686710989030614557826799

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