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Properties

Label 1-77-77.48-r1-0-0
Degree $1$
Conductor $77$
Sign $-0.999 + 0.0237i$
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  + (0.309 + 0.951i)2-s + (0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (−0.309 + 0.951i)6-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s − 10-s − 12-s + (−0.309 − 0.951i)13-s + (−0.809 + 0.587i)15-s + (0.309 − 0.951i)16-s + (−0.309 + 0.951i)17-s + (−0.809 + 0.587i)18-s + (0.809 + 0.587i)19-s + (−0.309 − 0.951i)20-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (−0.309 + 0.951i)6-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s − 10-s − 12-s + (−0.309 − 0.951i)13-s + (−0.809 + 0.587i)15-s + (0.309 − 0.951i)16-s + (−0.309 + 0.951i)17-s + (−0.809 + 0.587i)18-s + (0.809 + 0.587i)19-s + (−0.309 − 0.951i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.999 + 0.0237i$
Analytic conductor: \(8.27479\)
Root analytic conductor: \(8.27479\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (1:\ ),\ -0.999 + 0.0237i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02266707747 + 1.906796064i\)
\(L(\frac12)\) \(\approx\) \(0.02266707747 + 1.906796064i\)
\(L(1)\) \(\approx\) \(0.7741325218 + 1.120584305i\)
\(L(1)\) \(\approx\) \(0.7741325218 + 1.120584305i\)

Euler product

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

−30.69356985131694503817805008192, −29.3357084534566347298727659724, −28.69192617317322710232674195953, −27.34560646366463508626672654613, −26.36625278896538977575753259994, −24.70783622616225051262502257045, −24.05357829693473227426035056188, −22.88953755233245168231826368790, −21.33275456305101729363920313166, −20.50686681278753536104987344215, −19.64050162330897717789218179393, −18.76998089250467895054961547105, −17.47875976605526769592457628196, −15.7468934464535284623214729123, −14.30061170017889074399158013797, −13.38613517230271863066806552651, −12.38966346258742384208329110347, −11.419085604475855085886563457174, −9.43247116007288982370755704546, −8.84773330100784413449028946937, −7.21485128076092153274811121381, −5.14004297496793137250168033488, −3.80513127168214241401065811218, −2.270550432238876057037027632627, −0.80303374642404199096332552677, 2.94676864153906791016432740931, 4.01172656396526011895542430972, 5.6026071572628767382425131468, 7.24264279896595194018892750388, 8.12915303142387722894683161480, 9.52631903487203465586852420792, 10.811883255222403069494087736934, 12.744104739335567194070402430037, 14.00339062031564244178575087252, 14.95851788093919527411999530552, 15.52979869299744955409711100411, 16.894714620403733390182598824090, 18.2709995999917182762772920649, 19.37892806809055226231025218930, 20.799955806970688881927246580836, 22.106842935150130330964334955211, 22.67404585094512849781152939366, 24.144433049864158167490538165039, 25.26374887108828526912371305081, 26.12140946265374350972809224104, 26.92506026631125824631826832179, 27.74248605524994597476644252541, 29.84453729219671317641673916358, 30.90816518685042000342197671272, 31.51793102069988679050211046890

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