Properties

Label 8-135e4-1.1-c1e4-0-1
Degree $8$
Conductor $332150625$
Sign $1$
Analytic cond. $1.35034$
Root an. cond. $1.03825$
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  + 4·4-s + 4·16-s − 4·19-s + 8·25-s + 8·31-s + 10·49-s − 52·61-s − 16·64-s − 16·76-s + 20·79-s + 32·100-s + 32·109-s − 8·121-s + 32·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 34·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 2·4-s + 16-s − 0.917·19-s + 8/5·25-s + 1.43·31-s + 10/7·49-s − 6.65·61-s − 2·64-s − 1.83·76-s + 2.25·79-s + 16/5·100-s + 3.06·109-s − 0.727·121-s + 2.87·124-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.61·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1.35034\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{12} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.875805709\)
\(L(\frac12)\) \(\approx\) \(1.875805709\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
good2$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 64 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 125 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 164 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{4} \)
97$C_2^2$ \( ( 1 - 185 T^{2} + p^{2} T^{4} )^{2} \)
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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

−10.05893226594612080803856190738, −9.192806302771851579721409368435, −9.158001478947800703851831006164, −8.960327204148331412512532930119, −8.875409603854670975065563328688, −8.201017349929793109061297698646, −7.896057077059508667919054658929, −7.75017696126545425844621574372, −7.49608998997687987425970719676, −6.96175487110205739025560028789, −6.95737592293525677656342898956, −6.58304298289204776394057602223, −6.14522138017574112019349301678, −6.12665419268401875548806668363, −5.95031934677130676969642251552, −5.20419041329673760118541883973, −4.71949257514534425789498584027, −4.68698523821464655812134364779, −4.27148609216619931026052515335, −3.62318633195932765068653292906, −3.08130577747486131419126422490, −2.84906386063257071396066027475, −2.45660321617197241438548096896, −1.93981026467338891151854269245, −1.34607187220342934847295857564, 1.34607187220342934847295857564, 1.93981026467338891151854269245, 2.45660321617197241438548096896, 2.84906386063257071396066027475, 3.08130577747486131419126422490, 3.62318633195932765068653292906, 4.27148609216619931026052515335, 4.68698523821464655812134364779, 4.71949257514534425789498584027, 5.20419041329673760118541883973, 5.95031934677130676969642251552, 6.12665419268401875548806668363, 6.14522138017574112019349301678, 6.58304298289204776394057602223, 6.95737592293525677656342898956, 6.96175487110205739025560028789, 7.49608998997687987425970719676, 7.75017696126545425844621574372, 7.896057077059508667919054658929, 8.201017349929793109061297698646, 8.875409603854670975065563328688, 8.960327204148331412512532930119, 9.158001478947800703851831006164, 9.192806302771851579721409368435, 10.05893226594612080803856190738

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