Properties

Label 8-1216e4-1.1-c1e4-0-3
Degree $8$
Conductor $2.186\times 10^{12}$
Sign $1$
Analytic cond. $8888.79$
Root an. cond. $3.11605$
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  + 4·9-s − 16·13-s + 12·17-s + 14·25-s − 24·29-s + 8·37-s + 26·49-s + 24·53-s − 44·73-s − 6·81-s − 56·109-s − 64·117-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 48·153-s + 157-s + 163-s + 167-s + 108·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 4/3·9-s − 4.43·13-s + 2.91·17-s + 14/5·25-s − 4.45·29-s + 1.31·37-s + 26/7·49-s + 3.29·53-s − 5.14·73-s − 2/3·81-s − 5.36·109-s − 5.91·117-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 3.88·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(8888.79\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.190153847\)
\(L(\frac12)\) \(\approx\) \(1.190153847\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
good3$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{4} \)
23$C_2$ \( ( 1 - p T^{2} )^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 59 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
59$C_2$ \( ( 1 - p T^{2} )^{4} \)
61$C_2^2$ \( ( 1 - 119 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 166 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \)
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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

−7.09704410691534492595996752076, −7.07471507580215251000913315380, −6.58615030010862309697059409641, −6.44748727415321766850422115321, −5.73152534511079310640977110835, −5.70508764640493290913318046038, −5.66451625682008719978681713578, −5.31007649048149705749361548648, −5.19521999778825856942435381932, −5.17859800304580118990284867436, −4.68759101457801892524595668614, −4.29623557514844464208380390629, −4.22668199946644908781172766466, −4.01140617960797528362668159308, −3.95611627419880782726378646121, −3.30091327831325657875194798288, −3.09559554929016288658165323050, −2.81120755735681013721697386347, −2.59392046250544133744023485433, −2.37288885591025245132387812213, −2.09745859622874046655580551569, −1.60550461373467879002857087036, −1.17134323649567858376662904904, −1.01102998948017035303886863168, −0.24004515818610407882153837512, 0.24004515818610407882153837512, 1.01102998948017035303886863168, 1.17134323649567858376662904904, 1.60550461373467879002857087036, 2.09745859622874046655580551569, 2.37288885591025245132387812213, 2.59392046250544133744023485433, 2.81120755735681013721697386347, 3.09559554929016288658165323050, 3.30091327831325657875194798288, 3.95611627419880782726378646121, 4.01140617960797528362668159308, 4.22668199946644908781172766466, 4.29623557514844464208380390629, 4.68759101457801892524595668614, 5.17859800304580118990284867436, 5.19521999778825856942435381932, 5.31007649048149705749361548648, 5.66451625682008719978681713578, 5.70508764640493290913318046038, 5.73152534511079310640977110835, 6.44748727415321766850422115321, 6.58615030010862309697059409641, 7.07471507580215251000913315380, 7.09704410691534492595996752076

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