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Properties

Label 1-79-79.33-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: $\chi_{79} (33, \cdot )$
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

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

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