Properties

Label 8-700e4-1.1-c1e4-0-2
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 8·4-s − 4·7-s − 8·8-s + 16·14-s − 4·16-s − 4·23-s − 32·28-s + 20·29-s + 32·32-s − 44·43-s + 16·46-s − 2·49-s + 32·56-s − 80·58-s − 64·64-s + 12·67-s − 18·81-s + 176·86-s − 32·92-s + 8·98-s + 32·107-s + 60·109-s + 16·112-s + 160·116-s − 6·121-s + 127-s + ⋯
L(s)  = 1  − 2.82·2-s + 4·4-s − 1.51·7-s − 2.82·8-s + 4.27·14-s − 16-s − 0.834·23-s − 6.04·28-s + 3.71·29-s + 5.65·32-s − 6.70·43-s + 2.35·46-s − 2/7·49-s + 4.27·56-s − 10.5·58-s − 8·64-s + 1.46·67-s − 2·81-s + 18.9·86-s − 3.33·92-s + 0.808·98-s + 3.09·107-s + 5.74·109-s + 1.51·112-s + 14.8·116-s − 0.545·121-s + 0.0887·127-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

\(L(1)\) \(\approx\) \(0.08531349876\)
\(L(\frac12)\) \(\approx\) \(0.08531349876\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T + p T^{2} )^{2} \)
5 \( 1 \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
good3$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 3 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 11 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \)
59$C_2^2$ \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 3 T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 117 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 140 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 77 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 172 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 140 T^{2} + p^{2} T^{4} )^{2} \)
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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

−7.43682482600369517638195388443, −7.42641557209902586737057996713, −7.08684948045845682130025427804, −6.89070177980392820890008859170, −6.81720838421729738503968428847, −6.39548296638332241740208188831, −6.31628382333336540659434826476, −6.13500724509747292012278700101, −6.04746270182600918334889341751, −5.29208683329478640771215784511, −4.95334382961862985307434193233, −4.89857709992370256171778083501, −4.79964732481931439554506576631, −4.30397711727848389678941555694, −4.11424542553821051055063356186, −3.44616769979758323797794276976, −3.25674459130478828005607176175, −3.20676965909006776525020044636, −2.87589362058556018888713648461, −2.20112813303573228190374159614, −2.06685534072767990501392757166, −1.73819859360698334423726265906, −1.22102541568391849971256756268, −0.825728559347057657812503843887, −0.17987227981376109836873039345, 0.17987227981376109836873039345, 0.825728559347057657812503843887, 1.22102541568391849971256756268, 1.73819859360698334423726265906, 2.06685534072767990501392757166, 2.20112813303573228190374159614, 2.87589362058556018888713648461, 3.20676965909006776525020044636, 3.25674459130478828005607176175, 3.44616769979758323797794276976, 4.11424542553821051055063356186, 4.30397711727848389678941555694, 4.79964732481931439554506576631, 4.89857709992370256171778083501, 4.95334382961862985307434193233, 5.29208683329478640771215784511, 6.04746270182600918334889341751, 6.13500724509747292012278700101, 6.31628382333336540659434826476, 6.39548296638332241740208188831, 6.81720838421729738503968428847, 6.89070177980392820890008859170, 7.08684948045845682130025427804, 7.42641557209902586737057996713, 7.43682482600369517638195388443

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