Properties

Label 16-150e8-1.1-c1e8-0-2
Degree $16$
Conductor $2.563\times 10^{17}$
Sign $1$
Analytic cond. $4.23591$
Root an. cond. $1.09442$
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  − 2·16-s − 16·31-s + 112·61-s + 9·81-s − 20·121-s + 127-s + 131-s + 137-s + 139-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 + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  − 1/2·16-s − 2.87·31-s + 14.3·61-s + 81-s − 1.81·121-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 + 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 + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(4.23591\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{150} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.409898044\)
\(L(\frac12)\) \(\approx\) \(1.409898044\)
\(L(\frac{3}{2})\) 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 \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
5 \( 1 \)
good7 \( ( 1 - 94 T^{4} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 5 T^{2} + p^{2} T^{4} )^{4} \)
13 \( ( 1 + p^{2} T^{4} )^{4} \)
17 \( ( 1 + 47 T^{4} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{4} \)
23 \( ( 1 - 958 T^{4} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + p T^{2} )^{8} \)
31 \( ( 1 + 2 T + p T^{2} )^{8} \)
37 \( ( 1 + 1106 T^{4} + p^{4} T^{8} )^{2} \)
41 \( ( 1 - 55 T^{2} + p^{2} T^{4} )^{4} \)
43 \( ( 1 + 1778 T^{4} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + p^{2} T^{4} )^{4} \)
53 \( ( 1 - 718 T^{4} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \)
61 \( ( 1 - 14 T + p T^{2} )^{8} \)
67 \( ( 1 + 2471 T^{4} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - p T^{2} )^{8} \)
73 \( ( 1 - 5617 T^{4} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \)
83 \( ( 1 + 10871 T^{4} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{4} \)
97 \( ( 1 + 2498 T^{4} + p^{4} T^{8} )^{2} \)
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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

−6.05292294595787903370532722539, −5.65662929676451807619200184444, −5.51721419258324170294415894852, −5.50210182948814418269338715932, −5.46264046827006235950537540646, −5.29478349274904067171114416408, −5.14095023665045280523795529715, −4.94559894868430505510387522605, −4.80060521669202178668491220590, −4.59037963425345594477193943323, −4.08571258127475767880558726187, −4.00531687195082858702010787900, −3.87838820991488060311248225288, −3.75160830104294674073228019702, −3.74289328770649006938144316249, −3.61963447891937566463349296429, −3.23893213817240536143298662573, −2.70572854985074972695335552668, −2.62999409128695463453044170967, −2.47033492955680976088709044635, −2.21477996751888029364567173423, −1.96713013012517432355421949032, −1.82213642098685371348147253708, −1.05289566278120391432237533337, −0.77580983440891378740683602000, 0.77580983440891378740683602000, 1.05289566278120391432237533337, 1.82213642098685371348147253708, 1.96713013012517432355421949032, 2.21477996751888029364567173423, 2.47033492955680976088709044635, 2.62999409128695463453044170967, 2.70572854985074972695335552668, 3.23893213817240536143298662573, 3.61963447891937566463349296429, 3.74289328770649006938144316249, 3.75160830104294674073228019702, 3.87838820991488060311248225288, 4.00531687195082858702010787900, 4.08571258127475767880558726187, 4.59037963425345594477193943323, 4.80060521669202178668491220590, 4.94559894868430505510387522605, 5.14095023665045280523795529715, 5.29478349274904067171114416408, 5.46264046827006235950537540646, 5.50210182948814418269338715932, 5.51721419258324170294415894852, 5.65662929676451807619200184444, 6.05292294595787903370532722539

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

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