Properties

Label 8-1323e4-1.1-c1e4-0-0
Degree $8$
Conductor $3.064\times 10^{12}$
Sign $1$
Analytic cond. $12455.1$
Root an. cond. $3.25026$
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  − 2·4-s − 5·16-s − 20·25-s + 20·37-s − 28·43-s + 20·64-s − 4·67-s + 44·79-s + 40·100-s − 4·109-s + 4·121-s + 127-s + 131-s + 137-s + 139-s − 40·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 46·169-s + 56·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 4-s − 5/4·16-s − 4·25-s + 3.28·37-s − 4.26·43-s + 5/2·64-s − 0.488·67-s + 4.95·79-s + 4·100-s − 0.383·109-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.28·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.53·169-s + 4.26·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{12} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(12455.1\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1323} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{12} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1232486908\)
\(L(\frac12)\) \(\approx\) \(0.1232486908\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{4} \)
11$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
17$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 7 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 11 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 191 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.76091468600867781463162811670, −6.54944068419191107695854603579, −6.42416062462811241108957657743, −6.26322123447981412029234839099, −6.01439227984217810795751816635, −5.83732176664004834820286838236, −5.60660937413714552342491363907, −5.06192291845201827909245424734, −5.05479678264218516576894645230, −4.98434096540528908124536930641, −4.63597038455383774019261599012, −4.44878837514606012685335661438, −4.03302284170833088873553239871, −3.92153750015967194636978542834, −3.77611291281668921802120699173, −3.46179562049907954120705714476, −3.30728838117662858687280142171, −2.77611866617057690361176747986, −2.39766806998932388034598561324, −2.21746838255265038496659161753, −2.10839342389064504094631454996, −1.62016671749628625577025097916, −1.28489050705477328282958369640, −0.67060593351823556159654370372, −0.096030460006892425022648807939, 0.096030460006892425022648807939, 0.67060593351823556159654370372, 1.28489050705477328282958369640, 1.62016671749628625577025097916, 2.10839342389064504094631454996, 2.21746838255265038496659161753, 2.39766806998932388034598561324, 2.77611866617057690361176747986, 3.30728838117662858687280142171, 3.46179562049907954120705714476, 3.77611291281668921802120699173, 3.92153750015967194636978542834, 4.03302284170833088873553239871, 4.44878837514606012685335661438, 4.63597038455383774019261599012, 4.98434096540528908124536930641, 5.05479678264218516576894645230, 5.06192291845201827909245424734, 5.60660937413714552342491363907, 5.83732176664004834820286838236, 6.01439227984217810795751816635, 6.26322123447981412029234839099, 6.42416062462811241108957657743, 6.54944068419191107695854603579, 6.76091468600867781463162811670

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