Properties

Degree 2
Conductor $ 2^{7} $
Sign $1$
Motivic weight 3
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.92·3-s − 15.8·5-s + 17.8·7-s − 2.71·9-s + 52.9·11-s − 8.43·13-s + 78.1·15-s + 129.·17-s + 50.4·19-s − 87.9·21-s − 128.·23-s + 126.·25-s + 146.·27-s − 111.·29-s + 302.·31-s − 260.·33-s − 283.·35-s + 182.·37-s + 41.5·39-s − 94.5·41-s + 184.·43-s + 43.0·45-s − 296.·47-s − 24.1·49-s − 636.·51-s + 102.·53-s − 839.·55-s + ⋯
L(s)  = 1  − 0.948·3-s − 1.41·5-s + 0.964·7-s − 0.100·9-s + 1.45·11-s − 0.179·13-s + 1.34·15-s + 1.84·17-s + 0.609·19-s − 0.914·21-s − 1.16·23-s + 1.01·25-s + 1.04·27-s − 0.712·29-s + 1.75·31-s − 1.37·33-s − 1.36·35-s + 0.813·37-s + 0.170·39-s − 0.360·41-s + 0.654·43-s + 0.142·45-s − 0.921·47-s − 0.0704·49-s − 1.74·51-s + 0.266·53-s − 2.05·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(128\)    =    \(2^{7}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  $\chi_{128} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 128,\ (\ :3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(1.02101\)
\(L(\frac12)\)  \(\approx\)  \(1.02101\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 4.92T + 27T^{2} \)
5 \( 1 + 15.8T + 125T^{2} \)
7 \( 1 - 17.8T + 343T^{2} \)
11 \( 1 - 52.9T + 1.33e3T^{2} \)
13 \( 1 + 8.43T + 2.19e3T^{2} \)
17 \( 1 - 129.T + 4.91e3T^{2} \)
19 \( 1 - 50.4T + 6.85e3T^{2} \)
23 \( 1 + 128.T + 1.21e4T^{2} \)
29 \( 1 + 111.T + 2.43e4T^{2} \)
31 \( 1 - 302.T + 2.97e4T^{2} \)
37 \( 1 - 182.T + 5.06e4T^{2} \)
41 \( 1 + 94.5T + 6.89e4T^{2} \)
43 \( 1 - 184.T + 7.95e4T^{2} \)
47 \( 1 + 296.T + 1.03e5T^{2} \)
53 \( 1 - 102.T + 1.48e5T^{2} \)
59 \( 1 + 93.3T + 2.05e5T^{2} \)
61 \( 1 - 338.T + 2.26e5T^{2} \)
67 \( 1 - 489.T + 3.00e5T^{2} \)
71 \( 1 - 86.9T + 3.57e5T^{2} \)
73 \( 1 + 154.T + 3.89e5T^{2} \)
79 \( 1 - 449.T + 4.93e5T^{2} \)
83 \( 1 - 383.T + 5.71e5T^{2} \)
89 \( 1 + 517.T + 7.04e5T^{2} \)
97 \( 1 - 1.73e3T + 9.12e5T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.22145092644490242896657950484, −11.77829913500715860540719432261, −11.30207595140199057097919810845, −9.901445929724155115713989279880, −8.331248348067111324926838008591, −7.55580407382455872408433359252, −6.13768267618134977342485975481, −4.84312125171252364127696887047, −3.65871538204585852737982858969, −0.942123458980252545577404869939, 0.942123458980252545577404869939, 3.65871538204585852737982858969, 4.84312125171252364127696887047, 6.13768267618134977342485975481, 7.55580407382455872408433359252, 8.331248348067111324926838008591, 9.901445929724155115713989279880, 11.30207595140199057097919810845, 11.77829913500715860540719432261, 12.22145092644490242896657950484

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