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Properties

Label 1-77-77.37-r0-0-0
Degree $1$
Conductor $77$
Sign $0.0870 - 0.996i$
Analytic cond. $0.357586$
Root an. cond. $0.357586$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0870 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0870 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.0870 - 0.996i$
Analytic conductor: \(0.357586\)
Root analytic conductor: \(0.357586\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (0:\ ),\ 0.0870 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.077846772 - 0.9877809400i\)
\(L(\frac12)\) \(\approx\) \(1.077846772 - 0.9877809400i\)
\(L(1)\) \(\approx\) \(1.297672501 - 0.7790385360i\)
\(L(1)\) \(\approx\) \(1.297672501 - 0.7790385360i\)

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

−31.84905342887138728727433424263, −30.72860168909922521528719221633, −29.99619603267166464033858133474, −27.83054175733286694480871775958, −27.06201731281335474439170176930, −25.972045222160691265002976079322, −25.07355125292295264200875878384, −24.01854500385135383505823009013, −22.96548484887485251629940371728, −21.827754251923311386478910637480, −20.62040536756270822380014013154, −19.787527217514119985150389344490, −18.277219849377688680765484174112, −16.610941533829054281871391032615, −15.62712098900070918437734286691, −14.99149269834527046115497303196, −13.71155476539002874862799238174, −12.667652987561066263872993218632, −11.23139363422244992907978113385, −9.31262103810417548173007021334, −8.07903797534328757884524911726, −7.329053616237217400406805874116, −5.3100935802275824852196302679, −4.02306846432650952566367312208, −2.990833218132938635292311455936, 1.73665983393474109816296844380, 3.35077669443214817572836412607, 4.23041519628241824158797335031, 6.32440328551947295994073722629, 7.80262004878191632269846076261, 9.15494903348111520664321900691, 10.62782543526104499845491464518, 11.98040335166072241800330194065, 12.822538662864294086654898495178, 14.26216428819812681825326871651, 14.84755226907878755538677653623, 16.21661082359214147016086771515, 18.56110316881402305754620917904, 19.018797519589153132779611078500, 20.13521293333882290598786726675, 20.95237350028309256483508716156, 22.27023564981005092805027758795, 23.61375630475274514580503685340, 24.092785208774933087951198431969, 25.60955644085789686219251774722, 26.83276103672911397916842757374, 27.86858005659016004438590083136, 29.17589640226735494675897913518, 30.30660625033798915927631292054, 30.96776389807897477799050339202

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