Properties

Degree 2
Conductor 79
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 − 1.61·5-s − 8-s + 9-s − 1.00·10-s + 0.618·11-s + 0.618·13-s + 0.618·18-s − 1.61·19-s + 0.999·20-s + 0.381·22-s − 1.61·23-s + 1.61·25-s + 0.381·26-s + 0.618·31-s + 0.999·32-s − 0.618·36-s − 1.00·38-s + 1.61·40-s − 0.381·44-s − 1.61·45-s − 1.00·46-s + 49-s + 1.00·50-s − 0.381·52-s − 1.00·55-s + ⋯
L(s)  = 1  + 0.618·2-s − 0.618·4-s − 1.61·5-s − 8-s + 9-s − 1.00·10-s + 0.618·11-s + 0.618·13-s + 0.618·18-s − 1.61·19-s + 0.999·20-s + 0.381·22-s − 1.61·23-s + 1.61·25-s + 0.381·26-s + 0.618·31-s + 0.999·32-s − 0.618·36-s − 1.00·38-s + 1.61·40-s − 0.381·44-s − 1.61·45-s − 1.00·46-s + 49-s + 1.00·50-s − 0.381·52-s − 1.00·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(79\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{79} (78, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 79,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.5273458988\)
\(L(\frac12)\)  \(\approx\)  \(0.5273458988\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 79$,\(F_p(T)\) is a polynomial of degree 2. If $p = 79$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad79 \( 1 - T \)
good2 \( 1 - 0.618T + T^{2} \)
3 \( 1 - T^{2} \)
5 \( 1 + 1.61T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - 0.618T + T^{2} \)
13 \( 1 - 0.618T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.61T + T^{2} \)
23 \( 1 + 1.61T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 0.618T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.61T + T^{2} \)
83 \( 1 - 2T + T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 + 1.61T + T^{2} \)
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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

−14.79301903017583787602292860928, −13.55274723202347412692925946328, −12.47946133071320119957371003723, −11.81774686271880974772720205144, −10.41921929706756148110282082287, −8.873978905013962342505313744455, −7.87413839665017329527808259336, −6.38007356073625378647747108423, −4.39037048839329186211077735661, −3.84845369197390074372926468934, 3.84845369197390074372926468934, 4.39037048839329186211077735661, 6.38007356073625378647747108423, 7.87413839665017329527808259336, 8.873978905013962342505313744455, 10.41921929706756148110282082287, 11.81774686271880974772720205144, 12.47946133071320119957371003723, 13.55274723202347412692925946328, 14.79301903017583787602292860928

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