Properties

Label 16-280e8-1.1-c1e8-0-0
Degree $16$
Conductor $3.778\times 10^{19}$
Sign $1$
Analytic cond. $624.426$
Root an. cond. $1.49526$
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  − 8·2-s + 32·4-s − 80·8-s + 120·16-s − 24·23-s − 12·25-s − 32·32-s + 192·46-s + 96·50-s − 384·64-s − 32·71-s − 768·92-s − 384·100-s + 88·121-s + 127-s + 1.21e3·128-s + 131-s + 137-s + 139-s + 256·142-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + ⋯
L(s)  = 1  − 5.65·2-s + 16·4-s − 28.2·8-s + 30·16-s − 5.00·23-s − 2.39·25-s − 5.65·32-s + 28.3·46-s + 13.5·50-s − 48·64-s − 3.79·71-s − 80.0·92-s − 38.3·100-s + 8·121-s + 0.0887·127-s + 107.·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 21.4·142-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(624.426\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{280} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.04446252942\)
\(L(\frac12)\) \(\approx\) \(0.04446252942\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p T + p T^{2} )^{4} \)
5 \( 1 + 12 T^{2} + 72 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \)
7 \( ( 1 + p^{2} T^{4} )^{2} \)
good3 \( ( 1 - 4 T^{2} + 8 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )( 1 + 4 T^{2} + 8 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} ) \)
11 \( ( 1 - p T^{2} )^{8} \)
13 \( ( 1 - 36 T^{2} + 648 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )( 1 + 36 T^{2} + 648 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} ) \)
17 \( ( 1 + p^{2} T^{4} )^{4} \)
19 \( ( 1 - 12 T^{2} + 72 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 + 6 T + p T^{2} )^{4}( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
29 \( ( 1 + p T^{2} )^{8} \)
31 \( ( 1 - p T^{2} )^{8} \)
37 \( ( 1 + p^{2} T^{4} )^{4} \)
41 \( ( 1 - p T^{2} )^{8} \)
43 \( ( 1 + p^{2} T^{4} )^{4} \)
47 \( ( 1 + p^{2} T^{4} )^{4} \)
53 \( ( 1 + p^{2} T^{4} )^{4} \)
59 \( ( 1 + 108 T^{2} + 5832 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 12 T^{2} + 72 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( ( 1 + p^{2} T^{4} )^{4} \)
71 \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \)
73 \( ( 1 + p^{2} T^{4} )^{4} \)
79 \( ( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2}( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \)
83 \( ( 1 - 36 T^{2} + 648 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )( 1 + 36 T^{2} + 648 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} ) \)
89 \( ( 1 + p T^{2} )^{8} \)
97 \( ( 1 + p^{2} T^{4} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.65465557229243971316562137480, −5.40611655372319100474273376780, −4.85823112195092454801929660205, −4.82585642517749726050328896500, −4.67726531944294669460565997801, −4.55375349567762386340692504249, −4.45459309243281316717465369431, −4.15250423914634092087162403678, −4.08683563112174009025728619409, −4.08117541941384295050310182594, −3.62214230631845193078223354374, −3.60544078055219277810188060249, −3.37775816419388644356846843020, −2.90968454290696787704784180484, −2.73260854277117253896783117749, −2.70476761719547101851873728900, −2.19344225615377220546993600807, −2.08906694815753378939179253828, −1.87473191363303879610551512204, −1.76746752373120844314252423008, −1.70280530211943676702986266702, −1.61079898979287288665856417387, −0.889490430849294701351055520748, −0.42781277241968759865938235503, −0.40447727518961596242918048311, 0.40447727518961596242918048311, 0.42781277241968759865938235503, 0.889490430849294701351055520748, 1.61079898979287288665856417387, 1.70280530211943676702986266702, 1.76746752373120844314252423008, 1.87473191363303879610551512204, 2.08906694815753378939179253828, 2.19344225615377220546993600807, 2.70476761719547101851873728900, 2.73260854277117253896783117749, 2.90968454290696787704784180484, 3.37775816419388644356846843020, 3.60544078055219277810188060249, 3.62214230631845193078223354374, 4.08117541941384295050310182594, 4.08683563112174009025728619409, 4.15250423914634092087162403678, 4.45459309243281316717465369431, 4.55375349567762386340692504249, 4.67726531944294669460565997801, 4.82585642517749726050328896500, 4.85823112195092454801929660205, 5.40611655372319100474273376780, 5.65465557229243971316562137480

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

Plot not available for L-functions of degree greater than 10.