Properties

Label 8-384e4-1.1-c3e4-0-6
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $263505.$
Root an. cond. $4.75990$
Motivic weight $3$
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  + 50·9-s − 352·23-s − 292·25-s + 1.53e3·47-s + 1.16e3·49-s + 2.72e3·71-s + 1.68e3·73-s + 1.77e3·81-s − 4.24e3·97-s + 1.09e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.91e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 1.85·9-s − 3.19·23-s − 2.33·25-s + 4.76·47-s + 3.39·49-s + 4.54·71-s + 2.70·73-s + 2.42·81-s − 4.44·97-s + 0.820·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.23·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(263505.\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(5.439368101\)
\(L(\frac12)\) \(\approx\) \(5.439368101\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 - 50 T^{2} + p^{6} T^{4} \)
good5$C_2^2$ \( ( 1 + 146 T^{2} + p^{6} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 582 T^{2} + p^{6} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 546 T^{2} + p^{6} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 2458 T^{2} + p^{6} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 9410 T^{2} + p^{6} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 13614 T^{2} + p^{6} T^{4} )^{2} \)
23$C_2$ \( ( 1 + 88 T + p^{3} T^{2} )^{4} \)
29$C_2^2$ \( ( 1 - 16222 T^{2} + p^{6} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 13718 T^{2} + p^{6} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 8918 T^{2} + p^{6} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 101774 T^{2} + p^{6} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 103998 T^{2} + p^{6} T^{4} )^{2} \)
47$C_2$ \( ( 1 - 384 T + p^{3} T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + 87154 T^{2} + p^{6} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 13858 T^{2} + p^{6} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 398266 T^{2} + p^{6} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 598926 T^{2} + p^{6} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 680 T + p^{3} T^{2} )^{4} \)
73$C_2$ \( ( 1 - 422 T + p^{3} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 431862 T^{2} + p^{6} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 1108978 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 490994 T^{2} + p^{6} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 1062 T + p^{3} T^{2} )^{4} \)
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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

−7.82777149077445649775703971754, −7.73739291001308760953478211897, −7.19402619003018246735693884172, −6.91180106037657762344760500406, −6.89008274989190101928214915304, −6.73628133027435979123325019843, −6.02555236952748195748887724984, −5.94607300455247912007257813586, −5.85897032405063957481865082842, −5.51800786662508369692804379637, −5.16036968192028427278117327986, −5.05798927625358856048916462531, −4.18276300448995929055912623338, −4.11110490410458870102468278499, −4.07306887566258519281176068495, −4.03328660146504025408164781197, −3.69114182946198317984541354092, −3.10635570455659761780372494620, −2.43259681218300691228305394814, −2.20088709861418325556210129523, −2.13709432572039998663957880386, −1.80274762545815569547717434503, −1.09203255405296752864955148613, −0.71936775367820822958079707610, −0.44340141307865287955654488317, 0.44340141307865287955654488317, 0.71936775367820822958079707610, 1.09203255405296752864955148613, 1.80274762545815569547717434503, 2.13709432572039998663957880386, 2.20088709861418325556210129523, 2.43259681218300691228305394814, 3.10635570455659761780372494620, 3.69114182946198317984541354092, 4.03328660146504025408164781197, 4.07306887566258519281176068495, 4.11110490410458870102468278499, 4.18276300448995929055912623338, 5.05798927625358856048916462531, 5.16036968192028427278117327986, 5.51800786662508369692804379637, 5.85897032405063957481865082842, 5.94607300455247912007257813586, 6.02555236952748195748887724984, 6.73628133027435979123325019843, 6.89008274989190101928214915304, 6.91180106037657762344760500406, 7.19402619003018246735693884172, 7.73739291001308760953478211897, 7.82777149077445649775703971754

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