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Properties

Label 1-79-79.74-r1-0-0
Degree $1$
Conductor $79$
Sign $-0.471 - 0.881i$
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  + (0.428 + 0.903i)2-s + (−0.278 + 0.960i)3-s + (−0.632 + 0.774i)4-s + (−0.200 + 0.979i)5-s + (−0.987 + 0.160i)6-s + (−0.692 − 0.721i)7-s + (−0.970 − 0.239i)8-s + (−0.845 − 0.534i)9-s + (−0.970 + 0.239i)10-s + (0.948 + 0.316i)11-s + (−0.568 − 0.822i)12-s + (0.987 + 0.160i)13-s + (0.354 − 0.935i)14-s + (−0.885 − 0.464i)15-s + (−0.200 − 0.979i)16-s + (0.354 + 0.935i)17-s + ⋯
L(s)  = 1  + (0.428 + 0.903i)2-s + (−0.278 + 0.960i)3-s + (−0.632 + 0.774i)4-s + (−0.200 + 0.979i)5-s + (−0.987 + 0.160i)6-s + (−0.692 − 0.721i)7-s + (−0.970 − 0.239i)8-s + (−0.845 − 0.534i)9-s + (−0.970 + 0.239i)10-s + (0.948 + 0.316i)11-s + (−0.568 − 0.822i)12-s + (0.987 + 0.160i)13-s + (0.354 − 0.935i)14-s + (−0.885 − 0.464i)15-s + (−0.200 − 0.979i)16-s + (0.354 + 0.935i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $-0.471 - 0.881i$
Analytic conductor: \(8.48972\)
Root analytic conductor: \(8.48972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (1:\ ),\ -0.471 - 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5035966412 + 0.8405420241i\)
\(L(\frac12)\) \(\approx\) \(-0.5035966412 + 0.8405420241i\)
\(L(1)\) \(\approx\) \(0.4379298932 + 0.8016901402i\)
\(L(1)\) \(\approx\) \(0.4379298932 + 0.8016901402i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (0.428 + 0.903i)T \)
3 \( 1 + (-0.278 + 0.960i)T \)
5 \( 1 + (-0.200 + 0.979i)T \)
7 \( 1 + (-0.692 - 0.721i)T \)
11 \( 1 + (0.948 + 0.316i)T \)
13 \( 1 + (0.987 + 0.160i)T \)
17 \( 1 + (0.354 + 0.935i)T \)
19 \( 1 + (-0.919 + 0.391i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.0402 + 0.999i)T \)
31 \( 1 + (-0.996 - 0.0804i)T \)
37 \( 1 + (-0.799 - 0.600i)T \)
41 \( 1 + (0.748 + 0.663i)T \)
43 \( 1 + (-0.948 + 0.316i)T \)
47 \( 1 + (-0.799 + 0.600i)T \)
53 \( 1 + (-0.278 - 0.960i)T \)
59 \( 1 + (0.632 + 0.774i)T \)
61 \( 1 + (-0.120 + 0.992i)T \)
67 \( 1 + (0.568 + 0.822i)T \)
71 \( 1 + (0.970 + 0.239i)T \)
73 \( 1 + (0.987 - 0.160i)T \)
83 \( 1 + (-0.632 + 0.774i)T \)
89 \( 1 + (-0.970 + 0.239i)T \)
97 \( 1 + (0.120 - 0.992i)T \)
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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

−29.99440743653639347728388899274, −29.229587600205262504546664684273, −28.1677982816012207985943368509, −27.665557789220539851017632681250, −25.50062220565863158008859380085, −24.57412082994096424938913238569, −23.51797168990267026294523093100, −22.6483907503633651130208613092, −21.45880970173641082580363514910, −20.13533921723274056884086127093, −19.31587964244933348074523929240, −18.41140255455395092087945673100, −17.07081159495550423846524345476, −15.65331186021319605653556880139, −13.90381555469749202604396298158, −13.02461924159077799930973565558, −12.11110846518507209442778508431, −11.33081654707897630155992431525, −9.40427115937445843986967922529, −8.4509154035935753385179293291, −6.35567741875347516353105873708, −5.32345405224315748617730821659, −3.558752253572332851417252865504, −1.83113473618771564004108462040, −0.430108295714595453767374045752, 3.518518747404472420442161849342, 4.107287466367871831757731227256, 6.05216703752775878808110763326, 6.79378387022475794047606128342, 8.50779483288088645198912517451, 9.92649532576176715039928410459, 11.06401125471483028695299323456, 12.62357881963972771198060591126, 14.24242054343500573094958502745, 14.84540904577554405828082304229, 16.11919212893846903217706489214, 16.8510843895322844323536250198, 18.07637472632236875782539165019, 19.609525736043019325020621834189, 21.1270089552193425240677789075, 22.21659224823928472106560815131, 22.889984280985936502164927363287, 23.66776699257170081322621018287, 25.63176668921816605622303250720, 26.021253579438742603857123865793, 27.07299099248492172662315254249, 27.99688438910188657611069776416, 29.743960456666599777070233952446, 30.64599438334930338140123583330, 31.90957409357153918129934081622

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