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Properties

Label 1-79-79.41-r1-0-0
Degree $1$
Conductor $79$
Sign $-0.731 - 0.681i$
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  + (0.568 − 0.822i)2-s + (0.970 − 0.239i)3-s + (−0.354 − 0.935i)4-s + (−0.748 − 0.663i)5-s + (0.354 − 0.935i)6-s + (0.970 − 0.239i)7-s + (−0.970 − 0.239i)8-s + (0.885 − 0.464i)9-s + (−0.970 + 0.239i)10-s + (−0.748 + 0.663i)11-s + (−0.568 − 0.822i)12-s + (−0.354 − 0.935i)13-s + (0.354 − 0.935i)14-s + (−0.885 − 0.464i)15-s + (−0.748 + 0.663i)16-s + (0.354 + 0.935i)17-s + ⋯
L(s)  = 1  + (0.568 − 0.822i)2-s + (0.970 − 0.239i)3-s + (−0.354 − 0.935i)4-s + (−0.748 − 0.663i)5-s + (0.354 − 0.935i)6-s + (0.970 − 0.239i)7-s + (−0.970 − 0.239i)8-s + (0.885 − 0.464i)9-s + (−0.970 + 0.239i)10-s + (−0.748 + 0.663i)11-s + (−0.568 − 0.822i)12-s + (−0.354 − 0.935i)13-s + (0.354 − 0.935i)14-s + (−0.885 − 0.464i)15-s + (−0.748 + 0.663i)16-s + (0.354 + 0.935i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $-0.731 - 0.681i$
Analytic conductor: \(8.48972\)
Root analytic conductor: \(8.48972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (1:\ ),\ -0.731 - 0.681i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9570749362 - 2.430551930i\)
\(L(\frac12)\) \(\approx\) \(0.9570749362 - 2.430551930i\)
\(L(1)\) \(\approx\) \(1.252848639 - 1.211493445i\)
\(L(1)\) \(\approx\) \(1.252848639 - 1.211493445i\)

Euler product

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

−31.30685838622359787500013402153, −30.82929198405826418852477331456, −29.58178626589297208929144024795, −27.34468905611209932085390312635, −26.88428187546784723529379473108, −25.9402658432531069697328347576, −24.7174255663634459798489102785, −23.94796157123083239387048283168, −22.760376659475247482439470747988, −21.4227193431444558815820695694, −20.799587041253513429885908746699, −19.057641049905867387124684423241, −18.24179135531176781468611941786, −16.44378349206338874953369167623, −15.52470307020595686444430840854, −14.474821603056944684018912165483, −13.965927616796772539686345162110, −12.30850339093984561633550806178, −10.96883823642206165466848286136, −9.04613237111166187616432976044, −7.927566203851910689464659422, −7.17509929297274940301210765353, −5.19207283710109797934222449341, −3.909002236204302163528461375520, −2.66099821933606538305923253169, 1.03981985781569464145577073890, 2.58507846085268880178302459180, 4.083919835983646008430745429268, 5.11752018795900048714353757865, 7.50432881126653633930670635318, 8.52328054775459094688515988854, 9.95291672636795897246427191689, 11.31614419802810206052179918181, 12.65986423682485860558351369655, 13.30851376643930096895077892988, 14.91118618088928098122517246152, 15.30033982378301655719284854149, 17.506765151890583239539237297112, 18.805104929416479811462091386594, 19.87608665592587080976538876900, 20.54899295460264674028071118441, 21.32717666412018606864139204527, 23.02452177424059269732673752040, 23.996613485854560386549031152032, 24.68309065702242701804758929172, 26.35138797634316563568440565279, 27.51013564518792835438104171880, 28.25110541047192518550463666621, 29.78704130999339425552854314651, 30.6778801791682884220877263763

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