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Properties

Label 1-79-79.20-r0-0-0
Degree $1$
Conductor $79$
Sign $-0.524 - 0.851i$
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  + (−0.845 − 0.534i)2-s + (0.799 − 0.600i)3-s + (0.428 + 0.903i)4-s + (−0.632 − 0.774i)5-s + (−0.996 + 0.0804i)6-s + (−0.919 − 0.391i)7-s + (0.120 − 0.992i)8-s + (0.278 − 0.960i)9-s + (0.120 + 0.992i)10-s + (0.987 + 0.160i)11-s + (0.885 + 0.464i)12-s + (−0.996 − 0.0804i)13-s + (0.568 + 0.822i)14-s + (−0.970 − 0.239i)15-s + (−0.632 + 0.774i)16-s + (0.568 − 0.822i)17-s + ⋯
L(s)  = 1  + (−0.845 − 0.534i)2-s + (0.799 − 0.600i)3-s + (0.428 + 0.903i)4-s + (−0.632 − 0.774i)5-s + (−0.996 + 0.0804i)6-s + (−0.919 − 0.391i)7-s + (0.120 − 0.992i)8-s + (0.278 − 0.960i)9-s + (0.120 + 0.992i)10-s + (0.987 + 0.160i)11-s + (0.885 + 0.464i)12-s + (−0.996 − 0.0804i)13-s + (0.568 + 0.822i)14-s + (−0.970 − 0.239i)15-s + (−0.632 + 0.774i)16-s + (0.568 − 0.822i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $-0.524 - 0.851i$
Analytic conductor: \(0.366874\)
Root analytic conductor: \(0.366874\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (0:\ ),\ -0.524 - 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3308706521 - 0.5926381338i\)
\(L(\frac12)\) \(\approx\) \(0.3308706521 - 0.5926381338i\)
\(L(1)\) \(\approx\) \(0.6116901060 - 0.4579134219i\)
\(L(1)\) \(\approx\) \(0.6116901060 - 0.4579134219i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (-0.845 - 0.534i)T \)
3 \( 1 + (0.799 - 0.600i)T \)
5 \( 1 + (-0.632 - 0.774i)T \)
7 \( 1 + (-0.919 - 0.391i)T \)
11 \( 1 + (0.987 + 0.160i)T \)
13 \( 1 + (-0.996 - 0.0804i)T \)
17 \( 1 + (0.568 - 0.822i)T \)
19 \( 1 + (-0.200 - 0.979i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.692 - 0.721i)T \)
31 \( 1 + (-0.0402 + 0.999i)T \)
37 \( 1 + (0.948 + 0.316i)T \)
41 \( 1 + (-0.354 + 0.935i)T \)
43 \( 1 + (0.987 - 0.160i)T \)
47 \( 1 + (0.948 - 0.316i)T \)
53 \( 1 + (0.799 + 0.600i)T \)
59 \( 1 + (0.428 - 0.903i)T \)
61 \( 1 + (-0.748 + 0.663i)T \)
67 \( 1 + (0.885 + 0.464i)T \)
71 \( 1 + (0.120 - 0.992i)T \)
73 \( 1 + (-0.996 + 0.0804i)T \)
83 \( 1 + (0.428 + 0.903i)T \)
89 \( 1 + (0.120 + 0.992i)T \)
97 \( 1 + (-0.748 + 0.663i)T \)
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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

−31.850071784398395408247400709834, −30.44161875394966644992058622505, −29.241595127893433472803689650003, −27.77798507660705829776846091570, −27.123889031335672782820868114300, −26.15729705834132106457646944484, −25.43468167645141083676121113636, −24.35026092369577439085572088261, −22.80109098032286093581485872518, −21.849590306198758730960893123671, −20.112698510899545920722075133624, −19.314555881553682230640475058034, −18.71772555506358815387850557773, −16.88776153515973085444140046832, −16.00212742166100071145769071665, −14.85237652307074131967357227889, −14.339240007630710117628720190612, −12.170241246984810631010544393294, −10.54466393015390980174693467909, −9.705047709939625720626580508118, −8.52572402528889608765033800281, −7.35960897904582006997830481023, −6.076154805107898056555711329413, −3.959888852405097614285942414503, −2.46841791557211252641291032889, 0.962105647200944733433881559477, 2.803461622542368354302487948866, 4.08543116319098767110991918619, 6.85126653757022628234361513397, 7.73280697642879951356152423104, 9.06956824112963978816522073872, 9.75717254935971493118859088556, 11.78432511683228507478008700023, 12.49618834062193595636763449132, 13.667030006541177670434375748741, 15.41246942102641562528376675873, 16.595940476275401933055612016074, 17.65542642916796370790597109895, 19.21505480859568436324325849262, 19.68659640775683970729102606014, 20.36448999259427329887918772068, 21.823821662891264383839397519192, 23.37248890423415007594684416390, 24.73435518566220736926912129665, 25.41509238743166757682278226364, 26.647054691204425219558913603008, 27.45235612401552894678798480469, 28.74618975875157236394595787778, 29.68245374217950048096393447959, 30.52696399027471210586052649788

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