Properties

Label 8-700e4-1.1-c1e4-0-10
Degree $8$
Conductor $240100000000$
Sign $1$
Analytic cond. $976.114$
Root an. cond. $2.36421$
Motivic weight $1$
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 + 2·4-s + 4·8-s − 6·9-s + 8·16-s − 12·18-s + 24·19-s + 4·29-s − 24·31-s + 8·32-s − 12·36-s + 24·37-s + 48·38-s − 2·49-s − 8·53-s + 8·58-s − 48·62-s + 8·64-s − 24·72-s + 48·74-s + 48·76-s + 9·81-s − 48·83-s − 4·98-s + 48·103-s − 16·106-s + 36·109-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1.41·8-s − 2·9-s + 2·16-s − 2.82·18-s + 5.50·19-s + 0.742·29-s − 4.31·31-s + 1.41·32-s − 2·36-s + 3.94·37-s + 7.78·38-s − 2/7·49-s − 1.09·53-s + 1.05·58-s − 6.09·62-s + 64-s − 2.82·72-s + 5.57·74-s + 5.50·76-s + 81-s − 5.26·83-s − 0.404·98-s + 4.72·103-s − 1.55·106-s + 3.44·109-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: $1$
Analytic conductor: \(976.114\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{700} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.560562107\)
\(L(\frac12)\) \(\approx\) \(7.560562107\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
good3$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 30 T^{2} + 419 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$$\times$$C_2^2$ \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \)
17$D_4\times C_2$ \( 1 - 26 T^{2} + 315 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
23$D_4\times C_2$ \( 1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
37$D_{4}$ \( ( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 91 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
59$C_2^2$ \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 - 148 T^{2} + 11190 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
71$D_4\times C_2$ \( 1 - 228 T^{2} + 22310 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 124 T^{2} + 7590 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 94 T^{2} + 3891 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 260 T^{2} + 31014 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 - 154 T^{2} + 20859 T^{4} - 154 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.39167370676300212213757340421, −7.29151091134818957099096834831, −7.16543114526691949933733476125, −7.12696581110744306339585303740, −6.21501182523023839745274184099, −6.15113839267526925823601651602, −6.07742679421245999510993484189, −5.72154567469958886597099066419, −5.48761035990834622754531498287, −5.30854930963982759558531380329, −5.29633760129370767494228659319, −5.07353010652077893743457543866, −4.39834078297719984233271961274, −4.36969928787527045802651885516, −4.35005406136056641150195919797, −3.47421148196375176495463117074, −3.44792482444850013131517185154, −3.28058496071971325801247671847, −3.13490689453419752830703271468, −2.76817201527125476914216650411, −2.49704480360999234001562125660, −1.79747884005375589412942185917, −1.64720153014001527606567993930, −1.01518454152294140742474394079, −0.67420339793453416191755498985, 0.67420339793453416191755498985, 1.01518454152294140742474394079, 1.64720153014001527606567993930, 1.79747884005375589412942185917, 2.49704480360999234001562125660, 2.76817201527125476914216650411, 3.13490689453419752830703271468, 3.28058496071971325801247671847, 3.44792482444850013131517185154, 3.47421148196375176495463117074, 4.35005406136056641150195919797, 4.36969928787527045802651885516, 4.39834078297719984233271961274, 5.07353010652077893743457543866, 5.29633760129370767494228659319, 5.30854930963982759558531380329, 5.48761035990834622754531498287, 5.72154567469958886597099066419, 6.07742679421245999510993484189, 6.15113839267526925823601651602, 6.21501182523023839745274184099, 7.12696581110744306339585303740, 7.16543114526691949933733476125, 7.29151091134818957099096834831, 7.39167370676300212213757340421

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