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Properties

Label 1-67-67.63-r1-0-0
Degree $1$
Conductor $67$
Sign $0.988 - 0.148i$
Analytic cond. $7.20014$
Root an. cond. $7.20014$
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  + (−0.981 − 0.189i)2-s + (−0.415 − 0.909i)3-s + (0.928 + 0.371i)4-s + (0.959 − 0.281i)5-s + (0.235 + 0.971i)6-s + (0.327 + 0.945i)7-s + (−0.841 − 0.540i)8-s + (−0.654 + 0.755i)9-s + (−0.995 + 0.0950i)10-s + (−0.235 + 0.971i)11-s + (−0.0475 − 0.998i)12-s + (0.888 + 0.458i)13-s + (−0.142 − 0.989i)14-s + (−0.654 − 0.755i)15-s + (0.723 + 0.690i)16-s + (0.928 − 0.371i)17-s + ⋯
L(s)  = 1  + (−0.981 − 0.189i)2-s + (−0.415 − 0.909i)3-s + (0.928 + 0.371i)4-s + (0.959 − 0.281i)5-s + (0.235 + 0.971i)6-s + (0.327 + 0.945i)7-s + (−0.841 − 0.540i)8-s + (−0.654 + 0.755i)9-s + (−0.995 + 0.0950i)10-s + (−0.235 + 0.971i)11-s + (−0.0475 − 0.998i)12-s + (0.888 + 0.458i)13-s + (−0.142 − 0.989i)14-s + (−0.654 − 0.755i)15-s + (0.723 + 0.690i)16-s + (0.928 − 0.371i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $0.988 - 0.148i$
Analytic conductor: \(7.20014\)
Root analytic conductor: \(7.20014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (1:\ ),\ 0.988 - 0.148i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.096756122 - 0.08193442508i\)
\(L(\frac12)\) \(\approx\) \(1.096756122 - 0.08193442508i\)
\(L(1)\) \(\approx\) \(0.8030013764 - 0.1221846671i\)
\(L(1)\) \(\approx\) \(0.8030013764 - 0.1221846671i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (-0.981 - 0.189i)T \)
3 \( 1 + (-0.415 - 0.909i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
7 \( 1 + (0.327 + 0.945i)T \)
11 \( 1 + (-0.235 + 0.971i)T \)
13 \( 1 + (0.888 + 0.458i)T \)
17 \( 1 + (0.928 - 0.371i)T \)
19 \( 1 + (-0.327 + 0.945i)T \)
23 \( 1 + (0.580 - 0.814i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.888 - 0.458i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.786 + 0.618i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (-0.995 - 0.0950i)T \)
53 \( 1 + (0.142 + 0.989i)T \)
59 \( 1 + (0.841 + 0.540i)T \)
61 \( 1 + (-0.235 - 0.971i)T \)
71 \( 1 + (0.928 + 0.371i)T \)
73 \( 1 + (0.235 + 0.971i)T \)
79 \( 1 + (-0.0475 - 0.998i)T \)
83 \( 1 + (0.723 + 0.690i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−32.43392758313756951590399549576, −30.147116150934573246272209099893, −29.40781513827158099078434209777, −28.29682953013231206308643203575, −27.33779002734584784571796872206, −26.31928668029190457943976838043, −25.68347802605688731508912442219, −24.13603135003508945870617462494, −22.95942951852975252884987013691, −21.26361132357075040140972824342, −20.870523195457233990900965649724, −19.30446758147097438136739294262, −17.82829709594849816155371796141, −17.14975766340993596665184225053, −16.15651803831283717418196512524, −14.88926310381944032820329259058, −13.569628800172142311371031704264, −11.258810866427789505981574853145, −10.59020662116354083877634073968, −9.60169247136129671508218607621, −8.25929562046721596088698232123, −6.49793873432016503688532699815, −5.404064473070364014062235000143, −3.23079683932632607627412250191, −0.961123471375916943221645535359, 1.37188134997225650095504917346, 2.39391543814883779332314387935, 5.49855059942180627291505052858, 6.61662026408727582921717018814, 8.08188755165783856194729541667, 9.22443363111391359660270797307, 10.61712128554066959475031831806, 12.015033105227353632648810973062, 12.818202197632410426522380933819, 14.51764990926919211244427149538, 16.26447855959163843748469660898, 17.30837811981942718661299591915, 18.304367001476260303880027255245, 18.80531020649723601051120532262, 20.520110068826179547722883766460, 21.34926019232373652318813600379, 22.92729598341784675436916281454, 24.45502877455234818951806965511, 25.20598404988253349633175330786, 25.90130489242138796617331879568, 27.85288547826598060343474396905, 28.387127267290461495793961166897, 29.30631039934413056408603958949, 30.301104133249758987831685536175, 31.35429775254538066270783839885

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