Properties

Label 1-79-79.30-r1-0-0
Degree $1$
Conductor $79$
Sign $-0.667 + 0.744i$
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  + (−0.919 + 0.391i)2-s + (0.632 − 0.774i)3-s + (0.692 − 0.721i)4-s + (−0.0402 + 0.999i)5-s + (−0.278 + 0.960i)6-s + (−0.987 − 0.160i)7-s + (−0.354 + 0.935i)8-s + (−0.200 − 0.979i)9-s + (−0.354 − 0.935i)10-s + (−0.845 + 0.534i)11-s + (−0.120 − 0.992i)12-s + (0.278 + 0.960i)13-s + (0.970 − 0.239i)14-s + (0.748 + 0.663i)15-s + (−0.0402 − 0.999i)16-s + (0.970 + 0.239i)17-s + ⋯
L(s)  = 1  + (−0.919 + 0.391i)2-s + (0.632 − 0.774i)3-s + (0.692 − 0.721i)4-s + (−0.0402 + 0.999i)5-s + (−0.278 + 0.960i)6-s + (−0.987 − 0.160i)7-s + (−0.354 + 0.935i)8-s + (−0.200 − 0.979i)9-s + (−0.354 − 0.935i)10-s + (−0.845 + 0.534i)11-s + (−0.120 − 0.992i)12-s + (0.278 + 0.960i)13-s + (0.970 − 0.239i)14-s + (0.748 + 0.663i)15-s + (−0.0402 − 0.999i)16-s + (0.970 + 0.239i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $-0.667 + 0.744i$
Analytic conductor: \(8.48972\)
Root analytic conductor: \(8.48972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (1:\ ),\ -0.667 + 0.744i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2240452918 + 0.5018981870i\)
\(L(\frac12)\) \(\approx\) \(0.2240452918 + 0.5018981870i\)
\(L(1)\) \(\approx\) \(0.6223988943 + 0.1524084552i\)
\(L(1)\) \(\approx\) \(0.6223988943 + 0.1524084552i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (-0.919 + 0.391i)T \)
3 \( 1 + (0.632 - 0.774i)T \)
5 \( 1 + (-0.0402 + 0.999i)T \)
7 \( 1 + (-0.987 - 0.160i)T \)
11 \( 1 + (-0.845 + 0.534i)T \)
13 \( 1 + (0.278 + 0.960i)T \)
17 \( 1 + (0.970 + 0.239i)T \)
19 \( 1 + (-0.996 + 0.0804i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.948 + 0.316i)T \)
31 \( 1 + (0.799 + 0.600i)T \)
37 \( 1 + (-0.428 + 0.903i)T \)
41 \( 1 + (-0.885 + 0.464i)T \)
43 \( 1 + (0.845 + 0.534i)T \)
47 \( 1 + (-0.428 - 0.903i)T \)
53 \( 1 + (0.632 + 0.774i)T \)
59 \( 1 + (-0.692 - 0.721i)T \)
61 \( 1 + (-0.568 - 0.822i)T \)
67 \( 1 + (0.120 + 0.992i)T \)
71 \( 1 + (0.354 - 0.935i)T \)
73 \( 1 + (0.278 - 0.960i)T \)
83 \( 1 + (0.692 - 0.721i)T \)
89 \( 1 + (-0.354 - 0.935i)T \)
97 \( 1 + (0.568 + 0.822i)T \)
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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

−30.33312698682742850928070224920, −29.11843394879554617793219890136, −28.125714769132750197926930162415, −27.4631556321203814408411542868, −26.15340978277747899509512023358, −25.53922843262809031087462227219, −24.45403579160856096479175656987, −22.63534775371028283898610516638, −21.22830498316948477067568618387, −20.64374801142031887714347543873, −19.60689884626151795725917252345, −18.69506601402850364891367040220, −16.978751128723105275414836809682, −16.16890274728759033386371692009, −15.39804070309503314814008888612, −13.349014917555317218961801752159, −12.42127443733824031061475260068, −10.67277888180225591423109041140, −9.78494942273188896441817232531, −8.69780932428403701688834145613, −7.870642681763515302874401120649, −5.700782234257863227880996552816, −3.81048826503597000103626254713, −2.54971283596496024982127184848, −0.30898546586682104464500625590, 1.89748842145217392828857748873, 3.252773934254870768015161285015, 6.15495865702598995945369761152, 7.0013599274810898867803074459, 7.9882752593510202148866877069, 9.45529809321928318438308619617, 10.44066925961418162489474220214, 12.00166554872353840431101027486, 13.53582748344248632676057193543, 14.698556999575344789765941961573, 15.68214508531592239707832190602, 17.11802855626229649527025834648, 18.44304518503837746812837119525, 18.94139232312938578295320439849, 19.84637028096088940795184911994, 21.22076786363975658508434255147, 23.15741316662524489685270965373, 23.714699323419554211465621503951, 25.3448259881013093221337622640, 25.94151815566062482276877896851, 26.45504360026874976150528345574, 28.044725167764028870268503109623, 29.28198879489977474580161765070, 29.883353165287253943266547789757, 31.18782024429143671079914228262

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