Properties

Label 1-79-79.58-r1-0-0
Degree $1$
Conductor $79$
Sign $0.994 + 0.101i$
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  + (0.120 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.994 + 0.101i$
Analytic conductor: \(8.48972\)
Root analytic conductor: \(8.48972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (1:\ ),\ 0.994 + 0.101i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.079946150 + 0.1058330950i\)
\(L(\frac12)\) \(\approx\) \(2.079946150 + 0.1058330950i\)
\(L(1)\) \(\approx\) \(1.400108014 - 0.1173026790i\)
\(L(1)\) \(\approx\) \(1.400108014 - 0.1173026790i\)

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

−30.722561752589645195087400931712, −30.036579251538608846451709572646, −29.01530356990108260316549418491, −27.140305928858171154251094902550, −26.37541976695611423686919484587, −25.24781289243443264487857276450, −24.58696635270105904932961425820, −23.55496415765209684005509838262, −22.49657434261530626349680661171, −21.267059449985198584386986081535, −19.643060673105710854228841046858, −18.54688789346450627410625320356, −17.3777646737766204692172306652, −16.90973049687433369892205343540, −14.85660914331626707878886105038, −14.02227833678990806945694440840, −13.470018315807734794647049609387, −11.91051702146642858502126439699, −9.968307362046017103376985278049, −8.71803645456520929444943965876, −7.24672252533957715087313401956, −6.67716287746779968295884246595, −5.158650441189307244866219028441, −3.20343577177663372853064418164, −1.08615395631841161439111259890, 1.72837899927658469698956429773, 3.0858696643192084638248840189, 4.74181880226451311626491552834, 5.58371825053611653845529144763, 8.39562844241219515772241608974, 9.46734266033778409101367809817, 10.07329167597503480427953618290, 11.667672963493604680178285773381, 12.68451142734974393478439912888, 14.26043956361751811138520344257, 14.84143672855190877234677659823, 16.71210245984114065457863688494, 17.70244476290703643504682560344, 19.11936774154277788768360211476, 20.278818456407404943473785478472, 21.15133006927414015975526550762, 21.86561565893192986953839025575, 22.74634342722237550757444019747, 24.663841431135428355219592857032, 25.53347686761274989465654862022, 27.04379971931495359807239584602, 27.79429740574727733444471120317, 28.637896010706891349631753890506, 29.736030908497031659284100846012, 31.02495579096886406423752681885

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