Properties

Degree 2
Conductor 127
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61·2-s + 1.61·4-s − 8-s + 9-s + 0.618·11-s + 0.618·13-s − 1.61·17-s − 1.61·18-s − 1.61·19-s − 1.00·22-s + 25-s − 1.00·26-s − 1.61·31-s + 32-s + 2.61·34-s + 1.61·36-s − 1.61·37-s + 2.61·38-s + 0.618·41-s + 1.00·44-s + 0.618·47-s + 49-s − 1.61·50-s + 1.00·52-s − 1.61·61-s + 2.61·62-s − 1.61·64-s + ⋯
L(s)  = 1  − 1.61·2-s + 1.61·4-s − 8-s + 9-s + 0.618·11-s + 0.618·13-s − 1.61·17-s − 1.61·18-s − 1.61·19-s − 1.00·22-s + 25-s − 1.00·26-s − 1.61·31-s + 32-s + 2.61·34-s + 1.61·36-s − 1.61·37-s + 2.61·38-s + 0.618·41-s + 1.00·44-s + 0.618·47-s + 49-s − 1.61·50-s + 1.00·52-s − 1.61·61-s + 2.61·62-s − 1.61·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(127\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{127} (126, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 127,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.3276305465\)
\(L(\frac12)\)  \(\approx\)  \(0.3276305465\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 127$,\(F_p(T)\) is a polynomial of degree 2. If $p = 127$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad127 \( 1 - T \)
good2 \( 1 + 1.61T + T^{2} \)
3 \( 1 - T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - 0.618T + T^{2} \)
13 \( 1 - 0.618T + T^{2} \)
17 \( 1 + 1.61T + T^{2} \)
19 \( 1 + 1.61T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.61T + T^{2} \)
37 \( 1 + 1.61T + T^{2} \)
41 \( 1 - 0.618T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 0.618T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.61T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 1.61T + T^{2} \)
73 \( 1 - 0.618T + T^{2} \)
79 \( 1 - 0.618T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−13.47280047519811087717932057367, −12.42445559467085815708960642182, −10.94914769092000989049753263230, −10.55681981503439842271293523294, −9.134503101340332623126892173935, −8.692856212946880856357580576314, −7.24533615331990691556672324869, −6.48490417853134628055577876098, −4.26406518992861025057344194770, −1.85920683345632168750464233606, 1.85920683345632168750464233606, 4.26406518992861025057344194770, 6.48490417853134628055577876098, 7.24533615331990691556672324869, 8.692856212946880856357580576314, 9.134503101340332623126892173935, 10.55681981503439842271293523294, 10.94914769092000989049753263230, 12.42445559467085815708960642182, 13.47280047519811087717932057367

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