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Properties

Label 1-77-77.5-r1-0-0
Degree $1$
Conductor $77$
Sign $0.846 + 0.531i$
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  + (0.913 + 0.406i)2-s + (0.978 + 0.207i)3-s + (0.669 + 0.743i)4-s + (0.104 − 0.994i)5-s + (0.809 + 0.587i)6-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)12-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)15-s + (−0.104 + 0.994i)16-s + (−0.913 + 0.406i)17-s + (0.669 + 0.743i)18-s + (−0.669 + 0.743i)19-s + (0.809 − 0.587i)20-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)2-s + (0.978 + 0.207i)3-s + (0.669 + 0.743i)4-s + (0.104 − 0.994i)5-s + (0.809 + 0.587i)6-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)12-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)15-s + (−0.104 + 0.994i)16-s + (−0.913 + 0.406i)17-s + (0.669 + 0.743i)18-s + (−0.669 + 0.743i)19-s + (0.809 − 0.587i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.846 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.846 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.846 + 0.531i$
Analytic conductor: \(8.27479\)
Root analytic conductor: \(8.27479\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (1:\ ),\ 0.846 + 0.531i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.707079722 + 1.067047009i\)
\(L(\frac12)\) \(\approx\) \(3.707079722 + 1.067047009i\)
\(L(1)\) \(\approx\) \(2.378838054 + 0.5210483526i\)
\(L(1)\) \(\approx\) \(2.378838054 + 0.5210483526i\)

Euler product

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

−30.95505620835547289911173521442, −30.08531469315145059175579003969, −29.28548107801086078434728987798, −27.767181438701001538446517746052, −26.28764400755492143269720396431, −25.54267356758378540341055650689, −24.28809171987100100901214416426, −23.31070279086254573086961081308, −22.01629692392198359980468751160, −21.20398538312466158201110260169, −19.99166006578534245288252245691, −19.10576345518070547395180803443, −18.0803814165431516550776672157, −15.82455987681066403568822127455, −14.97194124282285670459479972756, −13.905027488978022653241144884947, −13.23562660141049302925392409861, −11.62433335356714352058577841791, −10.49800173263522739106886901838, −9.113325517930223781285768673202, −7.2726302515423533936326902705, −6.28880576306999885966609098040, −4.255648684434398441567611551369, −3.06093308805631416210404871689, −1.887469367094521218417312386065, 1.98763721747517250475928894696, 3.67810385437050222148008398633, 4.73383534523371336242153152375, 6.28001789782902715806102255992, 8.02540047258522027101920560376, 8.74367618180417541017233044750, 10.53426798834023786847120053708, 12.36471340356951890062723702426, 13.2094704303490453270506042679, 14.176387843959564624548868544, 15.45969298421012359980143632708, 16.188890529341750723810650337808, 17.5568410344470199935324777355, 19.42664301761620284626176158226, 20.58092437001950550125349547179, 21.044594125325704841553476960491, 22.363193820886357253598450362544, 23.74341884230657058574990075939, 24.689756368400975714206869106791, 25.40966391474158216010694318373, 26.46741783154146615104241641259, 27.82604568162710597888570337492, 29.19813128912832826136479215476, 30.49863797881627656926651674934, 31.199406700568886314776053723868

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