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Properties

Label 1-79-79.4-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: $\chi_{79} (4, \cdot )$
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

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

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