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Properties

Label 1-79-79.50-r0-0-0
Degree $1$
Conductor $79$
Sign $-0.773 - 0.633i$
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  + (−0.919 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $-0.773 - 0.633i$
Analytic conductor: \(0.366874\)
Root analytic conductor: \(0.366874\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (0:\ ),\ -0.773 - 0.633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1610406868 - 0.4508055810i\)
\(L(\frac12)\) \(\approx\) \(0.1610406868 - 0.4508055810i\)
\(L(1)\) \(\approx\) \(0.4454179739 - 0.3624612036i\)
\(L(1)\) \(\approx\) \(0.4454179739 - 0.3624612036i\)

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

−31.625557999958824128542484219031, −30.30530706846233359559938716827, −29.012925385292907228623493981380, −28.197183942518455294502435340205, −27.12330471749758309313703864492, −26.4862202395216315879465501820, −25.501813764309793615756647949079, −23.87869730276697427435150938360, −23.17873885378963778926395532548, −21.62820627479131549973998576519, −20.80194331875483444171150387985, −19.25884192252781913507506042542, −17.93316170291568632021363669259, −17.586010928587508155881696910431, −15.98528220312988154199400341068, −15.23397492950712923712171141761, −14.22030196559637954075574605970, −11.70773395710933031502834130476, −10.92840274256769238480134797360, −10.033939642397471933619083498526, −8.64066621789361655326997748227, −7.17286647346507897769641360352, −5.994066678795298252882136496991, −4.51054728367740388094681607844, −2.239268167154229217199349283681, 0.77663914887470648626135632770, 2.24600808969841665738679041368, 4.71146283627155183124251704609, 6.271631265245346849815181867719, 8.06445485484297199222014015906, 8.36479120058208516080603997722, 10.436665986424934322582498753289, 11.32065073489660406372517685481, 12.52292684808794998988014748238, 13.375478320452206725558971607994, 15.58549784841459813082381133363, 16.77404840643422925576006891599, 17.61453183589604509526587866180, 18.41382423574709993463487544981, 19.73695720783708517695422073340, 20.6754377282311765421762180017, 21.75723333602433817719216335374, 23.516144603990064756193895914431, 24.3888898850124567862532231262, 25.17209212203173064242741141463, 26.71304761576446602656716105778, 27.87358698287362347025494609830, 28.42140655917787093708256731299, 29.49563646194466610359689817167, 30.37667826849631467796935793797

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