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Properties

Label 1-77-77.74-r1-0-0
Degree $1$
Conductor $77$
Sign $0.303 + 0.952i$
Analytic cond. $8.27479$
Root an. cond. $8.27479$
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  + (−0.669 − 0.743i)2-s + (0.913 + 0.406i)3-s + (−0.104 + 0.994i)4-s + (−0.978 − 0.207i)5-s + (−0.309 − 0.951i)6-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + (0.5 + 0.866i)10-s + (−0.5 + 0.866i)12-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)15-s + (−0.978 − 0.207i)16-s + (−0.669 + 0.743i)17-s + (0.104 − 0.994i)18-s + (0.104 + 0.994i)19-s + (0.309 − 0.951i)20-s + ⋯
L(s)  = 1  + (−0.669 − 0.743i)2-s + (0.913 + 0.406i)3-s + (−0.104 + 0.994i)4-s + (−0.978 − 0.207i)5-s + (−0.309 − 0.951i)6-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + (0.5 + 0.866i)10-s + (−0.5 + 0.866i)12-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)15-s + (−0.978 − 0.207i)16-s + (−0.669 + 0.743i)17-s + (0.104 − 0.994i)18-s + (0.104 + 0.994i)19-s + (0.309 − 0.951i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.303 + 0.952i$
Analytic conductor: \(8.27479\)
Root analytic conductor: \(8.27479\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (1:\ ),\ 0.303 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7928479756 + 0.5797761440i\)
\(L(\frac12)\) \(\approx\) \(0.7928479756 + 0.5797761440i\)
\(L(1)\) \(\approx\) \(0.8211013048 + 0.06594312912i\)
\(L(1)\) \(\approx\) \(0.8211013048 + 0.06594312912i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.669 - 0.743i)T \)
3 \( 1 + (0.913 + 0.406i)T \)
5 \( 1 + (-0.978 - 0.207i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (-0.669 + 0.743i)T \)
19 \( 1 + (0.104 + 0.994i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (0.913 - 0.406i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.104 - 0.994i)T \)
53 \( 1 + (-0.978 + 0.207i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + (0.978 + 0.207i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.104 - 0.994i)T \)
79 \( 1 + (-0.669 - 0.743i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−30.9358749984154442080730571593, −29.82245419479349070562553025702, −28.381529513767545187615336186674, −27.11496730933831861469220514257, −26.58933955153761969488378424050, −25.41034994462262817810722207993, −24.476519644916477616108258796350, −23.62332735750600771368025761449, −22.39147159161146804328333278243, −20.26790112006270681117623106372, −19.75453245360620663351609732122, −18.62337278682156826704767915410, −17.7447400050455931622396812882, −16.03754718943321908166078067258, −15.24628303126628488663331775743, −14.295059613411615199208644851439, −12.91587323596839301210636094905, −11.246553968154707826081344973261, −9.74331546473643195374321013183, −8.4775289584696221283912251184, −7.64172586394826919537503488031, −6.608497143955168168463948665144, −4.56743437280845701743562319239, −2.69356026270233266803972238867, −0.54138558364351828696127675316, 1.81695266124800131173800294883, 3.47331796367863675772417375771, 4.38647334471547579119703519842, 7.28453675039893634592049072428, 8.31765251726373417617990605451, 9.26169698053530304472236409980, 10.54187310197458927121001253411, 11.783481284240019302951210843888, 12.96645474762810963162742044019, 14.416254036510179789040760224997, 15.80537915925150585660308154646, 16.6934108884377241721220161038, 18.37832294698605624840881493047, 19.47865125851700053316589241593, 19.96622634016099908100353435515, 21.16507035102140867614207196232, 22.07336619399538721633362815333, 23.7011875653759174960256279003, 25.07144864402516504758648468351, 26.23708518706473594277534784411, 26.98143663390834772853279921599, 27.79122571942636409026580420289, 28.94432995464327089082348404978, 30.315001712051873248789964427664, 31.2448917499912842329795282608

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