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Properties

Label 1-79-79.64-r0-0-0
Degree $1$
Conductor $79$
Sign $-0.949 + 0.315i$
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  + (0.120 + 0.992i)2-s + (−0.354 + 0.935i)3-s + (−0.970 + 0.239i)4-s + (0.885 + 0.464i)5-s + (−0.970 − 0.239i)6-s + (−0.354 + 0.935i)7-s + (−0.354 − 0.935i)8-s + (−0.748 − 0.663i)9-s + (−0.354 + 0.935i)10-s + (0.885 − 0.464i)11-s + (0.120 − 0.992i)12-s + (−0.970 + 0.239i)13-s + (−0.970 − 0.239i)14-s + (−0.748 + 0.663i)15-s + (0.885 − 0.464i)16-s + (−0.970 + 0.239i)17-s + ⋯
L(s)  = 1  + (0.120 + 0.992i)2-s + (−0.354 + 0.935i)3-s + (−0.970 + 0.239i)4-s + (0.885 + 0.464i)5-s + (−0.970 − 0.239i)6-s + (−0.354 + 0.935i)7-s + (−0.354 − 0.935i)8-s + (−0.748 − 0.663i)9-s + (−0.354 + 0.935i)10-s + (0.885 − 0.464i)11-s + (0.120 − 0.992i)12-s + (−0.970 + 0.239i)13-s + (−0.970 − 0.239i)14-s + (−0.748 + 0.663i)15-s + (0.885 − 0.464i)16-s + (−0.970 + 0.239i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $-0.949 + 0.315i$
Analytic conductor: \(0.366874\)
Root analytic conductor: \(0.366874\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (0:\ ),\ -0.949 + 0.315i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1378259159 + 0.8524498354i\)
\(L(\frac12)\) \(\approx\) \(0.1378259159 + 0.8524498354i\)
\(L(1)\) \(\approx\) \(0.5565926711 + 0.7658838861i\)
\(L(1)\) \(\approx\) \(0.5565926711 + 0.7658838861i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (0.120 + 0.992i)T \)
3 \( 1 + (-0.354 + 0.935i)T \)
5 \( 1 + (0.885 + 0.464i)T \)
7 \( 1 + (-0.354 + 0.935i)T \)
11 \( 1 + (0.885 - 0.464i)T \)
13 \( 1 + (-0.970 + 0.239i)T \)
17 \( 1 + (-0.970 + 0.239i)T \)
19 \( 1 + (0.568 - 0.822i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.748 + 0.663i)T \)
31 \( 1 + (0.120 + 0.992i)T \)
37 \( 1 + (0.568 - 0.822i)T \)
41 \( 1 + (0.885 + 0.464i)T \)
43 \( 1 + (0.885 + 0.464i)T \)
47 \( 1 + (0.568 + 0.822i)T \)
53 \( 1 + (-0.354 - 0.935i)T \)
59 \( 1 + (-0.970 - 0.239i)T \)
61 \( 1 + (0.568 - 0.822i)T \)
67 \( 1 + (0.120 - 0.992i)T \)
71 \( 1 + (-0.354 - 0.935i)T \)
73 \( 1 + (-0.970 - 0.239i)T \)
83 \( 1 + (-0.970 + 0.239i)T \)
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

−30.31608178064287125133747817480, −29.40630859839146369393127595969, −29.05140654233411885749980073339, −27.816199417001282187424522035824, −26.54693491298436871132146186702, −25.06385269626050898863292070935, −24.14277356333891336254911995948, −22.76485397740998006034064252999, −22.21082089760911315502685024992, −20.55416999352780106923222502214, −19.871091569203036187109185027057, −18.71464001250797174298866237569, −17.35081791050024711164856515868, −17.044667369296569336660706295480, −14.4215455288568767265271966536, −13.44646282531703793777322107031, −12.70032920691473200448618935518, −11.55759763688766257990765630302, −10.17968409076963973669293451770, −9.127481045788776902757746238233, −7.34714990212176345633339091218, −5.85864409117864126366930397907, −4.42509625607770214753796008598, −2.42703264411533900450807665187, −1.08172433191368472619260241554, 3.04001919889095820538526986714, 4.77514649264613067909647481449, 5.86123439837617545205031949010, 6.821329754974677782877705560098, 9.10368888951863127891315286169, 9.38648039379869883957713053689, 11.09880235259536146896770631693, 12.69307481341677982531122086063, 14.21850719719916006318200571292, 14.985297236013550582421792670848, 16.08421267045185981702805940712, 17.176461711268151963821134051681, 17.96154754954301085079789596519, 19.42681079111936021886699454943, 21.44804414878087805925746245913, 22.02783565813621359582164585949, 22.6310386738382907429482350899, 24.32636265307706311813529740483, 25.20496600104113113521825458144, 26.2968276371191200860658871080, 27.0236477464875621294931395433, 28.26674323909585712204314417482, 29.25634258025086841260474689890, 30.89282850063824424490033360090, 32.04602044861392219002459963285

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