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Properties

Label 1-79-79.12-r1-0-0
Degree $1$
Conductor $79$
Sign $-0.0352 + 0.999i$
Analytic cond. $8.48972$
Root an. cond. $8.48972$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0352 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0352 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $-0.0352 + 0.999i$
Analytic conductor: \(8.48972\)
Root analytic conductor: \(8.48972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (1:\ ),\ -0.0352 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.161057838 + 1.202738435i\)
\(L(\frac12)\) \(\approx\) \(1.161057838 + 1.202738435i\)
\(L(1)\) \(\approx\) \(0.9927766231 + 0.5396901488i\)
\(L(1)\) \(\approx\) \(0.9927766231 + 0.5396901488i\)

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

−30.428767120363118561677426265357, −29.47838054590995748611960394617, −28.515060935457270164307909018503, −27.36188399862872431498898664666, −26.290475721736044684109384975870, −25.17073810591700052805767964825, −24.654454173959034692029122900138, −23.40866916400959136115366168739, −21.20311993427555666660198606763, −20.54097051828617041097817267723, −19.79581936677741070112985470472, −18.41274755160603777233620910745, −17.56182102973102048851461073939, −16.60320733713093066809323685659, −14.979844128741046318795632542983, −13.633312811766014618868201977785, −12.534975891165003319562611377094, −11.23078575231346559574376782663, −9.610407138397664762055785360861, −8.761737234374616021265586312425, −7.66406843232046682948395046538, −6.43764900751400334086158768455, −4.108084873518886039817749911734, −2.03808324790370842941118811983, −1.14578931781675366093990646637, 1.86863369515931499689503053434, 3.19000829776914269234329835661, 5.44855782023524566621705843652, 6.864384600022234648654792364677, 8.52249610725597565836072018881, 9.04765511701590252394348077700, 10.63894441996271783497889731159, 11.18570135943159099008788155662, 13.58949285129576314251760380417, 14.84612702724785866817426686000, 15.407707925110990618314872453331, 16.86794393160026910427669287245, 18.07221917920969311174590582421, 18.95516548358792496053010164211, 20.11524118052616919069533492215, 21.30435918429786638043018044082, 22.05427431066096697712733503769, 24.04351532505097133116054343385, 25.189608955790032485697087147067, 25.79317666449015933084938816987, 26.918588472910780923638618168877, 27.570253298577476400687856309134, 28.7596853930210382616515854878, 30.13566469815525178417014188110, 30.93142255324120397308351535856

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