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Properties

Label 1-61-61.45-r0-0-0
Degree $1$
Conductor $61$
Sign $-0.645 - 0.763i$
Analytic cond. $0.283282$
Root an. cond. $0.283282$
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 + (−0.809 + 0.587i)3-s + (0.669 + 0.743i)4-s + (−0.978 + 0.207i)5-s + (0.978 − 0.207i)6-s + (0.104 − 0.994i)7-s + (−0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (0.978 + 0.207i)10-s − 11-s + (−0.978 − 0.207i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (0.669 − 0.743i)15-s + (−0.104 + 0.994i)16-s + (−0.669 − 0.743i)17-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)2-s + (−0.809 + 0.587i)3-s + (0.669 + 0.743i)4-s + (−0.978 + 0.207i)5-s + (0.978 − 0.207i)6-s + (0.104 − 0.994i)7-s + (−0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (0.978 + 0.207i)10-s − 11-s + (−0.978 − 0.207i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (0.669 − 0.743i)15-s + (−0.104 + 0.994i)16-s + (−0.669 − 0.743i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $-0.645 - 0.763i$
Analytic conductor: \(0.283282\)
Root analytic conductor: \(0.283282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (0:\ ),\ -0.645 - 0.763i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08093649629 - 0.1743856903i\)
\(L(\frac12)\) \(\approx\) \(0.08093649629 - 0.1743856903i\)
\(L(1)\) \(\approx\) \(0.3531059027 - 0.09540506537i\)
\(L(1)\) \(\approx\) \(0.3531059027 - 0.09540506537i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 \)
good2 \( 1 + (-0.913 - 0.406i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.978 + 0.207i)T \)
7 \( 1 + (0.104 - 0.994i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.669 + 0.743i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (-0.913 - 0.406i)T \)
67 \( 1 + (0.978 - 0.207i)T \)
71 \( 1 + (0.978 + 0.207i)T \)
73 \( 1 + (-0.978 - 0.207i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (0.913 + 0.406i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (0.913 - 0.406i)T \)
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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

−33.47767784622159384105058127905, −31.74778012513779994388714176410, −30.71721899269738905823053323647, −29.056580529803295721104731876280, −28.49554763389291302332113142207, −27.53541372631374314232845616959, −26.417810506093973286209221866767, −24.94489544025353185776790447477, −24.04840869916998445286072236699, −23.320413222136277317909952335, −21.72658509948815190013627781105, −20.02323123173963492266336906051, −18.77884136798761715485110392097, −18.32814591509732773791083496189, −16.79204212366000459809303358784, −15.96653436160049556400634605655, −14.787471536231545175582968397872, −12.59909755863699520617811461268, −11.64561340366547336647033883866, −10.50689342402669108161292091718, −8.66429405315832046720274032761, −7.67219534346604537564900568980, −6.34694822468116499583391022342, −4.98631501144302845724997580884, −2.05305769506285976718642232447, 0.33665885582553542356903238814, 3.20684990608160820099375396219, 4.70890926690848682224796984325, 6.92146355343710013921995220195, 7.953930956520024617791460172879, 9.76252372409691448360364675270, 10.81057888038671348940521218813, 11.53460310400407502297760427148, 12.97359673179230077703644137103, 15.34845615060353517268329256021, 16.05917780596691886296671319594, 17.32623450114381689417105440853, 18.18931465850253614584016174867, 19.7453419015734378211729459987, 20.51528655527737608202876314424, 21.88140771369841721501595519620, 23.099616704735279293011623344015, 24.11430357096802568689897478216, 26.02466340882899363847479405385, 26.95922060197971329463304960137, 27.46957598575859255707710174246, 28.70655287914566912096511732671, 29.6329058357830077877005338289, 30.73860321041861002079339183899, 32.199762734787638296729841353368

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