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Properties

Label 8-605e4-1.1-c1e4-0-8
Degree $8$
Conductor $133974300625$
Sign $1$
Analytic cond. $544.665$
Root an. cond. $2.19794$
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  + 12·9-s − 3·16-s + 10·25-s − 16·59-s + 32·71-s + 90·81-s + 127-s + 131-s + 137-s + 139-s − 36·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 120·225-s + 227-s + ⋯
L(s)  = 1  + 4·9-s − 3/4·16-s + 2·25-s − 2.08·59-s + 3.79·71-s + 10·81-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 8·225-s + 0.0663·227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(544.665\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.957440765\)
\(L(\frac12)\) \(\approx\) \(4.957440765\)
\(L(\frac{3}{2})\) 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)$
bad5$C_2$ \( ( 1 - p T^{2} )^{2} \)
11 \( 1 \)
good2$D_4\times C_2$ \( 1 + 3 T^{4} + p^{4} T^{8} \)
3$C_2$ \( ( 1 - p T^{2} )^{4} \)
7$D_4\times C_2$ \( 1 + 78 T^{4} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 - 162 T^{4} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 402 T^{4} + p^{4} T^{8} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2$ \( ( 1 - p T^{2} )^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{4} \)
43$D_4\times C_2$ \( 1 - 3522 T^{4} + p^{4} T^{8} \)
47$C_2$ \( ( 1 - p T^{2} )^{4} \)
53$C_2$ \( ( 1 - p T^{2} )^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
61$C_2$ \( ( 1 + p T^{2} )^{4} \)
67$C_2$ \( ( 1 - p T^{2} )^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
73$D_4\times C_2$ \( 1 + 10638 T^{4} + p^{4} T^{8} \)
79$C_2$ \( ( 1 + p T^{2} )^{4} \)
83$D_4\times C_2$ \( 1 - 13602 T^{4} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - p T^{2} )^{4} \)
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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

−7.58493797868114238595442546196, −7.40588690468398587926988855619, −7.29148769442485065294208979565, −6.98047121238297238110909992043, −6.69245220103188484435876803412, −6.47523948457421079185322869497, −6.45618254281969152852327513068, −6.32188019999766908839703632204, −5.75836689030499676939960622157, −5.19503777080631978638760048124, −5.09364302244767252011136695758, −5.05998464086956486955126877504, −4.70626903179474012636819906792, −4.34959750042247961506898204308, −4.21503985637338129807853813363, −3.90201486553940857797504435134, −3.87882076135721165959321579650, −3.23112208064544733888896798367, −3.08236766931578157297296305224, −2.65372624563422364466919908071, −2.00658937323569381458871732384, −1.93086453900231116528494668270, −1.61141406837569076504724973586, −0.953751842303099005902313268726, −0.859393174673052303896204801828, 0.859393174673052303896204801828, 0.953751842303099005902313268726, 1.61141406837569076504724973586, 1.93086453900231116528494668270, 2.00658937323569381458871732384, 2.65372624563422364466919908071, 3.08236766931578157297296305224, 3.23112208064544733888896798367, 3.87882076135721165959321579650, 3.90201486553940857797504435134, 4.21503985637338129807853813363, 4.34959750042247961506898204308, 4.70626903179474012636819906792, 5.05998464086956486955126877504, 5.09364302244767252011136695758, 5.19503777080631978638760048124, 5.75836689030499676939960622157, 6.32188019999766908839703632204, 6.45618254281969152852327513068, 6.47523948457421079185322869497, 6.69245220103188484435876803412, 6.98047121238297238110909992043, 7.29148769442485065294208979565, 7.40588690468398587926988855619, 7.58493797868114238595442546196

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