Properties

Label 8-2352e4-1.1-c1e4-0-15
Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Analytic cond. $124410.$
Root an. cond. $4.33368$
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  − 2·3-s + 9-s + 4·19-s − 4·25-s + 2·27-s + 24·29-s − 16·31-s − 8·37-s + 24·47-s + 12·53-s − 8·57-s − 24·59-s + 8·75-s − 4·81-s − 48·87-s + 32·93-s + 32·103-s − 4·109-s + 16·111-s + 24·113-s − 16·121-s + 127-s + 131-s + 137-s + 139-s − 48·141-s + 149-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s + 0.917·19-s − 4/5·25-s + 0.384·27-s + 4.45·29-s − 2.87·31-s − 1.31·37-s + 3.50·47-s + 1.64·53-s − 1.05·57-s − 3.12·59-s + 0.923·75-s − 4/9·81-s − 5.14·87-s + 3.31·93-s + 3.15·103-s − 0.383·109-s + 1.51·111-s + 2.25·113-s − 1.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.04·141-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(124410.\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.872545498\)
\(L(\frac12)\) \(\approx\) \(2.872545498\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( ( 1 + T + T^{2} )^{2} \)
7 \( 1 \)
good5$C_2^3$ \( 1 + 4 T^{2} - 9 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^3$ \( 1 + 16 T^{2} + 135 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^3$ \( 1 + 28 T^{2} + 495 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 8 T^{2} - 465 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 28 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^3$ \( 1 - 70 T^{2} - 429 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^3$ \( 1 + 62 T^{2} - 2397 T^{4} + 62 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2$ \( ( 1 + p T^{2} )^{4} \)
89$C_2^3$ \( 1 + 28 T^{2} - 7137 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.27248433950940626256940116751, −6.02142150210284893618772497087, −5.97130361136082310649151070030, −5.89901052118215065652730917486, −5.54738202302543604243309939454, −5.24873053526900987029210012738, −5.20323457421702508154381541519, −5.01982271692768564683111061404, −4.81203620857356592707589930764, −4.42482339946714923429809616059, −4.34843140861437959207839131815, −4.07764025658447903415538303607, −3.84839612273862822053438899270, −3.66834958311122374618159206961, −3.25949690413410416749106224637, −2.94275885217016444434900147809, −2.92415285928886926240643224666, −2.74419680636911592601699580804, −2.19878250973835774707589691550, −1.96622288449934021402527546144, −1.74370731683984079919659315943, −1.38036711545011244182255370376, −0.913523190910416646209209140266, −0.59667187597707975811514369782, −0.47921110023694565691298562825, 0.47921110023694565691298562825, 0.59667187597707975811514369782, 0.913523190910416646209209140266, 1.38036711545011244182255370376, 1.74370731683984079919659315943, 1.96622288449934021402527546144, 2.19878250973835774707589691550, 2.74419680636911592601699580804, 2.92415285928886926240643224666, 2.94275885217016444434900147809, 3.25949690413410416749106224637, 3.66834958311122374618159206961, 3.84839612273862822053438899270, 4.07764025658447903415538303607, 4.34843140861437959207839131815, 4.42482339946714923429809616059, 4.81203620857356592707589930764, 5.01982271692768564683111061404, 5.20323457421702508154381541519, 5.24873053526900987029210012738, 5.54738202302543604243309939454, 5.89901052118215065652730917486, 5.97130361136082310649151070030, 6.02142150210284893618772497087, 6.27248433950940626256940116751

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