Properties

Label 16-690e8-1.1-c1e8-0-1
Degree $16$
Conductor $5.138\times 10^{22}$
Sign $1$
Analytic cond. $849199.$
Root an. cond. $2.34727$
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  − 8·3-s + 32·9-s + 8·13-s − 2·16-s − 20·25-s − 88·27-s + 32·31-s + 16·37-s − 64·39-s − 48·43-s + 16·48-s + 64·61-s + 16·67-s − 8·73-s + 160·75-s + 206·81-s − 256·93-s − 16·97-s − 64·103-s − 128·111-s + 256·117-s + 56·121-s + 127-s + 384·129-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 4.61·3-s + 32/3·9-s + 2.21·13-s − 1/2·16-s − 4·25-s − 16.9·27-s + 5.74·31-s + 2.63·37-s − 10.2·39-s − 7.31·43-s + 2.30·48-s + 8.19·61-s + 1.95·67-s − 0.936·73-s + 18.4·75-s + 22.8·81-s − 26.5·93-s − 1.62·97-s − 6.30·103-s − 12.1·111-s + 23.6·117-s + 5.09·121-s + 0.0887·127-s + 33.8·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 23^{8}\)
Sign: $1$
Analytic conductor: \(849199.\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{690} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 23^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3073858926\)
\(L(\frac12)\) \(\approx\) \(0.3073858926\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{4} )^{2} \)
3 \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
5 \( ( 1 + p T^{2} )^{4} \)
23 \( ( 1 + T^{4} )^{2} \)
good7 \( ( 1 + p^{2} T^{4} )^{4} \)
11 \( ( 1 - 6 T + p T^{2} )^{4}( 1 + 6 T + p T^{2} )^{4} \)
13 \( ( 1 - 4 T + 8 T^{2} + 20 T^{3} - 274 T^{4} + 20 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( 1 - 36 T^{4} - 88634 T^{8} - 36 p^{4} T^{12} + p^{8} T^{16} \)
19 \( ( 1 - 48 T^{2} + 1138 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \)
31 \( ( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \)
37 \( ( 1 - 8 T + 32 T^{2} - 200 T^{3} + 1106 T^{4} - 200 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( ( 1 + 12 T^{2} + 838 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 + 24 T + 288 T^{2} + 2280 T^{3} + 15346 T^{4} + 2280 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 + 3332 T^{4} + 5069958 T^{8} + 3332 p^{4} T^{12} + p^{8} T^{16} \)
53 \( ( 1 - 5362 T^{4} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{4} \)
61 \( ( 1 - 8 T + p T^{2} )^{8} \)
67 \( ( 1 - 8 T + 32 T^{2} - 440 T^{3} + 5906 T^{4} - 440 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - 240 T^{2} + 24322 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + 4 T + 8 T^{2} - 20 T^{3} - 6034 T^{4} - 20 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 - 148 T^{2} + p^{2} T^{4} )^{4} \)
83 \( 1 + 3492 T^{4} + 59862118 T^{8} + 3492 p^{4} T^{12} + p^{8} T^{16} \)
89 \( ( 1 + 52 T^{2} - 6522 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 + 8 T + 32 T^{2} + 680 T^{3} + 14306 T^{4} + 680 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−4.65728336252795099705528580465, −4.33099613638906673084004542123, −4.24095318563335927402088242370, −4.21148002782277551353241201141, −4.13733923143049356280263859201, −3.98192286443537433050914471755, −3.85035966592552600181741963429, −3.83423178494857693586391526688, −3.65766338503335376828317630412, −3.23993514281653328672569629859, −3.10980569070032016261842038759, −3.00489455149613466222323655977, −2.88662618544675750812313032048, −2.77402710710490542156853599727, −2.28486817663019095808799361808, −2.18574697193643455854648703239, −2.07113355285060370684453634066, −1.73710811464688145499964739068, −1.65352922810162116145508389694, −1.62283979579580855374254019429, −0.958778701376155870771859703289, −0.884448797860282078120196194643, −0.872111323977647044875938444791, −0.59463492846513305198195169951, −0.19384124299261953454612110680, 0.19384124299261953454612110680, 0.59463492846513305198195169951, 0.872111323977647044875938444791, 0.884448797860282078120196194643, 0.958778701376155870771859703289, 1.62283979579580855374254019429, 1.65352922810162116145508389694, 1.73710811464688145499964739068, 2.07113355285060370684453634066, 2.18574697193643455854648703239, 2.28486817663019095808799361808, 2.77402710710490542156853599727, 2.88662618544675750812313032048, 3.00489455149613466222323655977, 3.10980569070032016261842038759, 3.23993514281653328672569629859, 3.65766338503335376828317630412, 3.83423178494857693586391526688, 3.85035966592552600181741963429, 3.98192286443537433050914471755, 4.13733923143049356280263859201, 4.21148002782277551353241201141, 4.24095318563335927402088242370, 4.33099613638906673084004542123, 4.65728336252795099705528580465

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

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