Properties

Label 1-79-79.67-r0-0-0
Degree $1$
Conductor $79$
Sign $0.787 - 0.616i$
Analytic cond. $0.366874$
Root an. cond. $0.366874$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.970 + 0.239i)2-s + (−0.748 − 0.663i)3-s + (0.885 − 0.464i)4-s + (0.568 + 0.822i)5-s + (0.885 + 0.464i)6-s + (−0.748 − 0.663i)7-s + (−0.748 + 0.663i)8-s + (0.120 + 0.992i)9-s + (−0.748 − 0.663i)10-s + (0.568 − 0.822i)11-s + (−0.970 − 0.239i)12-s + (0.885 − 0.464i)13-s + (0.885 + 0.464i)14-s + (0.120 − 0.992i)15-s + (0.568 − 0.822i)16-s + (0.885 − 0.464i)17-s + ⋯
L(s)  = 1  + (−0.970 + 0.239i)2-s + (−0.748 − 0.663i)3-s + (0.885 − 0.464i)4-s + (0.568 + 0.822i)5-s + (0.885 + 0.464i)6-s + (−0.748 − 0.663i)7-s + (−0.748 + 0.663i)8-s + (0.120 + 0.992i)9-s + (−0.748 − 0.663i)10-s + (0.568 − 0.822i)11-s + (−0.970 − 0.239i)12-s + (0.885 − 0.464i)13-s + (0.885 + 0.464i)14-s + (0.120 − 0.992i)15-s + (0.568 − 0.822i)16-s + (0.885 − 0.464i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.787 - 0.616i$
Analytic conductor: \(0.366874\)
Root analytic conductor: \(0.366874\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (0:\ ),\ 0.787 - 0.616i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5085425018 - 0.1753052999i\)
\(L(\frac12)\) \(\approx\) \(0.5085425018 - 0.1753052999i\)
\(L(1)\) \(\approx\) \(0.6052266510 - 0.09269378688i\)
\(L(1)\) \(\approx\) \(0.6052266510 - 0.09269378688i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (-0.970 + 0.239i)T \)
3 \( 1 + (-0.748 - 0.663i)T \)
5 \( 1 + (0.568 + 0.822i)T \)
7 \( 1 + (-0.748 - 0.663i)T \)
11 \( 1 + (0.568 - 0.822i)T \)
13 \( 1 + (0.885 - 0.464i)T \)
17 \( 1 + (0.885 - 0.464i)T \)
19 \( 1 + (-0.354 - 0.935i)T \)
23 \( 1 + T \)
29 \( 1 + (0.120 - 0.992i)T \)
31 \( 1 + (-0.970 + 0.239i)T \)
37 \( 1 + (-0.354 - 0.935i)T \)
41 \( 1 + (0.568 + 0.822i)T \)
43 \( 1 + (0.568 + 0.822i)T \)
47 \( 1 + (-0.354 + 0.935i)T \)
53 \( 1 + (-0.748 + 0.663i)T \)
59 \( 1 + (0.885 + 0.464i)T \)
61 \( 1 + (-0.354 - 0.935i)T \)
67 \( 1 + (-0.970 - 0.239i)T \)
71 \( 1 + (-0.748 + 0.663i)T \)
73 \( 1 + (0.885 + 0.464i)T \)
83 \( 1 + (0.885 - 0.464i)T \)
89 \( 1 + (-0.748 - 0.663i)T \)
97 \( 1 + (-0.354 - 0.935i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−31.22933225729110014720019257625, −29.58531398910177808152339074243, −28.8331934342464461451974731163, −28.050436221598819533115636193340, −27.42092297269630660417255030398, −25.75206930985426951420787197151, −25.332592204313806927217172380980, −23.75741921554143313687850764878, −22.343760829846651406518156448943, −21.21944455361436354205297322962, −20.54867731684138783536450608103, −19.06291207083484259463162818784, −17.91993745585884623552289440317, −16.79508310932586102854958504767, −16.28059695179844830481031298681, −14.98999722982031773301975261525, −12.69481532966258394713597795141, −12.00579280568597825702971768465, −10.50028193430843738505224823974, −9.49943376295343005608311632273, −8.75363409411150233726233382474, −6.6595760841322623850766245666, −5.57983625219725862907960885599, −3.69603675810424868601351865513, −1.5334427964803433990184099762, 1.04423942968282711145887922015, 2.96596485578092600739786149357, 5.82217146185363144523620512185, 6.58432894249349101289545692016, 7.60359781690986182138875524587, 9.32672900597427416977493533230, 10.658823281814912486087625030799, 11.27664193169556932614436230516, 13.053043506473191813521728442208, 14.24233824630328632652721032507, 15.94253241313451308137918870427, 16.9171106324099628817439764861, 17.803168857495536430537063886091, 18.83862836342336518846847790715, 19.533238751923913976470172885659, 21.21695154926155592309913850085, 22.65375078668766803960499921199, 23.487363311598451349269006971630, 24.85784126736185355960852316715, 25.64915039991913065153482261242, 26.723483007137815456333475062337, 27.85001686980563806882513009540, 29.00379156733842163939754299411, 29.761357550138103111772888901674, 30.3078400011827043236824584104

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