Properties

Label 16-768e8-1.1-c2e8-0-3
Degree $16$
Conductor $1.210\times 10^{23}$
Sign $1$
Analytic cond. $3.67764\times 10^{10}$
Root an. cond. $4.57454$
Motivic weight $2$
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  + 12·9-s − 168·25-s + 40·49-s + 240·73-s − 54·81-s + 720·97-s − 104·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.20e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 2.01e3·225-s + ⋯
L(s)  = 1  + 4/3·9-s − 6.71·25-s + 0.816·49-s + 3.28·73-s − 2/3·81-s + 7.42·97-s − 0.859·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 7.14·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s − 8.95·225-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{64} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(3.67764\times 10^{10}\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{64} \cdot 3^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.810541642\)
\(L(\frac12)\) \(\approx\) \(1.810541642\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 - 2 p T^{2} + p^{4} T^{4} )^{2} \)
good5 \( ( 1 + 42 T^{2} + p^{4} T^{4} )^{4} \)
7 \( ( 1 - 10 T^{2} + p^{4} T^{4} )^{4} \)
11 \( ( 1 + 26 T^{2} + p^{4} T^{4} )^{4} \)
13 \( ( 1 - 302 T^{2} + p^{4} T^{4} )^{4} \)
17 \( ( 1 - 66 T^{2} + p^{4} T^{4} )^{4} \)
19 \( ( 1 - 614 T^{2} + p^{4} T^{4} )^{4} \)
23 \( ( 1 - 194 T^{2} + p^{4} T^{4} )^{4} \)
29 \( ( 1 + 714 T^{2} + p^{4} T^{4} )^{4} \)
31 \( ( 1 + 950 T^{2} + p^{4} T^{4} )^{4} \)
37 \( ( 1 - 1294 T^{2} + p^{4} T^{4} )^{4} \)
41 \( ( 1 - 3330 T^{2} + p^{4} T^{4} )^{4} \)
43 \( ( 1 - 3590 T^{2} + p^{4} T^{4} )^{4} \)
47 \( ( 1 - 962 T^{2} + p^{4} T^{4} )^{4} \)
53 \( ( 1 + 5418 T^{2} + p^{4} T^{4} )^{4} \)
59 \( ( 1 + 6746 T^{2} + p^{4} T^{4} )^{4} \)
61 \( ( 1 - 120 T + p^{2} T^{2} )^{4}( 1 + 120 T + p^{2} T^{2} )^{4} \)
67 \( ( 1 + 4090 T^{2} + p^{4} T^{4} )^{4} \)
71 \( ( 1 - 9218 T^{2} + p^{4} T^{4} )^{4} \)
73 \( ( 1 - 30 T + p^{2} T^{2} )^{8} \)
79 \( ( 1 + 11510 T^{2} + p^{4} T^{4} )^{4} \)
83 \( ( 1 + 8378 T^{2} + p^{4} T^{4} )^{4} \)
89 \( ( 1 - 15810 T^{2} + p^{4} T^{4} )^{4} \)
97 \( ( 1 - 90 T + p^{2} T^{2} )^{8} \)
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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

−4.07015011691127770312581768667, −3.97195849902344313582987669162, −3.96180888093498413793600099780, −3.84231964929535587753167942784, −3.81261319620451432218785841852, −3.75730725928778319106569187902, −3.57046650509870409998582967347, −3.12044220026226644210272365448, −3.06059251670852986094368675326, −3.04113696329850970147931218443, −2.91369737425859011357488861201, −2.77628400062833147855616378093, −2.17288242567227121422561621114, −2.16903798266262465411547342446, −2.15957567560509698437862531929, −1.94142990280540867928825417132, −1.89949099977462808961943591563, −1.83987324392800606400921710170, −1.65196693554788972264992883234, −1.33699225142950291166865298341, −0.886798325364409416727438719638, −0.837020041185984528994816062138, −0.68544944918677965716472106940, −0.39669365840226956002399520981, −0.11726133449850119240940581440, 0.11726133449850119240940581440, 0.39669365840226956002399520981, 0.68544944918677965716472106940, 0.837020041185984528994816062138, 0.886798325364409416727438719638, 1.33699225142950291166865298341, 1.65196693554788972264992883234, 1.83987324392800606400921710170, 1.89949099977462808961943591563, 1.94142990280540867928825417132, 2.15957567560509698437862531929, 2.16903798266262465411547342446, 2.17288242567227121422561621114, 2.77628400062833147855616378093, 2.91369737425859011357488861201, 3.04113696329850970147931218443, 3.06059251670852986094368675326, 3.12044220026226644210272365448, 3.57046650509870409998582967347, 3.75730725928778319106569187902, 3.81261319620451432218785841852, 3.84231964929535587753167942784, 3.96180888093498413793600099780, 3.97195849902344313582987669162, 4.07015011691127770312581768667

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

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