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  + 0.618·2-s − 0.618·4-s − 8-s + 9-s − 1.61·11-s − 1.61·13-s + 0.618·17-s + 0.618·18-s + 0.618·19-s − 1.00·22-s + 25-s − 1.00·26-s + 0.618·31-s + 0.999·32-s + 0.381·34-s − 0.618·36-s + 0.618·37-s + 0.381·38-s − 1.61·41-s + 0.999·44-s − 1.61·47-s + 49-s + 0.618·50-s + 0.999·52-s + 0.618·61-s + 0.381·62-s + 0.618·64-s + ⋯
L(s)  = 1  + 0.618·2-s − 0.618·4-s − 8-s + 9-s − 1.61·11-s − 1.61·13-s + 0.618·17-s + 0.618·18-s + 0.618·19-s − 1.00·22-s + 25-s − 1.00·26-s + 0.618·31-s + 0.999·32-s + 0.381·34-s − 0.618·36-s + 0.618·37-s + 0.381·38-s − 1.61·41-s + 0.999·44-s − 1.61·47-s + 49-s + 0.618·50-s + 0.999·52-s + 0.618·61-s + 0.381·62-s + 0.618·64-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  =  \((2,\ 127,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.6772697748\)
\(L(\frac12)\)  \(\approx\)  \(0.6772697748\)
\(L(1)\)   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 \( 1 - 0.618T + T^{2} \)
3 \( 1 - T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + 1.61T + T^{2} \)
13 \( 1 + 1.61T + T^{2} \)
17 \( 1 - 0.618T + T^{2} \)
19 \( 1 - 0.618T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 0.618T + T^{2} \)
37 \( 1 - 0.618T + T^{2} \)
41 \( 1 + 1.61T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 1.61T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 0.618T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 0.618T + T^{2} \)
73 \( 1 + 1.61T + T^{2} \)
79 \( 1 + 1.61T + T^{2} \)
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

−13.42370588620668565955718275798, −12.77787074675390494649691571006, −11.96568531091756821059460171339, −10.26636958165125741525006942691, −9.697520639371775939113980261152, −8.173263553916733694561930400661, −7.11502962858507382676754181005, −5.35328852554235248733096561064, −4.64627465124050951229319361935, −2.93544618436142802891119222963, 2.93544618436142802891119222963, 4.64627465124050951229319361935, 5.35328852554235248733096561064, 7.11502962858507382676754181005, 8.173263553916733694561930400661, 9.697520639371775939113980261152, 10.26636958165125741525006942691, 11.96568531091756821059460171339, 12.77787074675390494649691571006, 13.42370588620668565955718275798

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