Properties

Degree $8$
Conductor $240100000000$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 6·3-s + 2·4-s + 12·6-s − 8·7-s − 4·8-s + 15·9-s − 12·12-s + 16·14-s + 8·16-s − 30·18-s + 48·21-s − 2·23-s + 24·24-s − 18·27-s − 16·28-s − 16·29-s − 8·32-s + 30·36-s − 96·42-s + 8·43-s + 4·46-s − 30·47-s − 48·48-s + 34·49-s + 36·54-s + 32·56-s + ⋯
L(s)  = 1  − 1.41·2-s − 3.46·3-s + 4-s + 4.89·6-s − 3.02·7-s − 1.41·8-s + 5·9-s − 3.46·12-s + 4.27·14-s + 2·16-s − 7.07·18-s + 10.4·21-s − 0.417·23-s + 4.89·24-s − 3.46·27-s − 3.02·28-s − 2.97·29-s − 1.41·32-s + 5·36-s − 14.8·42-s + 1.21·43-s + 0.589·46-s − 4.37·47-s − 6.92·48-s + 34/7·49-s + 4.89·54-s + 4.27·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{700} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{8} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5 \( 1 \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \)
11$C_2^3$ \( 1 + 21 T^{2} + 320 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^3$ \( 1 - 31 T^{2} + 672 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$$\times$$C_2^2$ \( ( 1 - 37 T^{2} + p^{2} T^{4} )( 1 + 26 T^{2} + p^{2} T^{4} ) \)
23$C_2^2$ \( ( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
31$C_2^2$$\times$$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )( 1 - 13 T^{2} + p^{2} T^{4} ) \)
37$C_2^3$ \( 1 + 65 T^{2} + 2856 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 15 T + 122 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \)
53$C_2^3$ \( 1 + 105 T^{2} + 8216 T^{4} + 105 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^3$ \( 1 - 91 T^{2} + 4800 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^3$ \( 1 - 71 T^{2} - 288 T^{4} - 71 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^3$ \( 1 + 77 T^{2} - 312 T^{4} + 77 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 27 T + 332 T^{2} + 27 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \)
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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

−7.917279309222902455252546946437, −7.67051299293385130172972427734, −7.47457579605821980622061543019, −7.07831323330829588749992574098, −6.91643814560194720328877919890, −6.60719548535132736608479386710, −6.52493700935694254359418641056, −6.42619331226204030803229455757, −6.00802259076958612267629136481, −5.93843318049884808087865252854, −5.77036011178632091281173706890, −5.69160011151600175782804437617, −5.16839598387022676127077856254, −5.14450797898659285188970422895, −5.04493490332622481562196194867, −4.11056799359014177845823238096, −4.06016488768825871883223120311, −3.98288927177179315572824353925, −3.46529966640841022350420741100, −2.95585148792669214423101273942, −2.89891023692161438738605574849, −2.82535537886469655401796563754, −2.00556098058206946839174090192, −1.35875752573773810669244344562, −1.24570098572474117884266241376, 0, 0, 0, 0, 1.24570098572474117884266241376, 1.35875752573773810669244344562, 2.00556098058206946839174090192, 2.82535537886469655401796563754, 2.89891023692161438738605574849, 2.95585148792669214423101273942, 3.46529966640841022350420741100, 3.98288927177179315572824353925, 4.06016488768825871883223120311, 4.11056799359014177845823238096, 5.04493490332622481562196194867, 5.14450797898659285188970422895, 5.16839598387022676127077856254, 5.69160011151600175782804437617, 5.77036011178632091281173706890, 5.93843318049884808087865252854, 6.00802259076958612267629136481, 6.42619331226204030803229455757, 6.52493700935694254359418641056, 6.60719548535132736608479386710, 6.91643814560194720328877919890, 7.07831323330829588749992574098, 7.47457579605821980622061543019, 7.67051299293385130172972427734, 7.917279309222902455252546946437

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