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Properties

Label 1-67-67.31-r1-0-0
Degree $1$
Conductor $67$
Sign $-0.0427 + 0.999i$
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  + (−0.235 − 0.971i)2-s + (0.142 − 0.989i)3-s + (−0.888 + 0.458i)4-s + (−0.415 − 0.909i)5-s + (−0.995 + 0.0950i)6-s + (−0.723 + 0.690i)7-s + (0.654 + 0.755i)8-s + (−0.959 − 0.281i)9-s + (−0.786 + 0.618i)10-s + (0.995 + 0.0950i)11-s + (0.327 + 0.945i)12-s + (−0.981 − 0.189i)13-s + (0.841 + 0.540i)14-s + (−0.959 + 0.281i)15-s + (0.580 − 0.814i)16-s + (−0.888 − 0.458i)17-s + ⋯
L(s)  = 1  + (−0.235 − 0.971i)2-s + (0.142 − 0.989i)3-s + (−0.888 + 0.458i)4-s + (−0.415 − 0.909i)5-s + (−0.995 + 0.0950i)6-s + (−0.723 + 0.690i)7-s + (0.654 + 0.755i)8-s + (−0.959 − 0.281i)9-s + (−0.786 + 0.618i)10-s + (0.995 + 0.0950i)11-s + (0.327 + 0.945i)12-s + (−0.981 − 0.189i)13-s + (0.841 + 0.540i)14-s + (−0.959 + 0.281i)15-s + (0.580 − 0.814i)16-s + (−0.888 − 0.458i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0427 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0427 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $-0.0427 + 0.999i$
Analytic conductor: \(7.20014\)
Root analytic conductor: \(7.20014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (1:\ ),\ -0.0427 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2311552968 - 0.2412487050i\)
\(L(\frac12)\) \(\approx\) \(-0.2311552968 - 0.2412487050i\)
\(L(1)\) \(\approx\) \(0.3840667339 - 0.4842414348i\)
\(L(1)\) \(\approx\) \(0.3840667339 - 0.4842414348i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (-0.235 - 0.971i)T \)
3 \( 1 + (0.142 - 0.989i)T \)
5 \( 1 + (-0.415 - 0.909i)T \)
7 \( 1 + (-0.723 + 0.690i)T \)
11 \( 1 + (0.995 + 0.0950i)T \)
13 \( 1 + (-0.981 - 0.189i)T \)
17 \( 1 + (-0.888 - 0.458i)T \)
19 \( 1 + (0.723 + 0.690i)T \)
23 \( 1 + (0.928 - 0.371i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.981 + 0.189i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.0475 - 0.998i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + (-0.786 - 0.618i)T \)
53 \( 1 + (-0.841 - 0.540i)T \)
59 \( 1 + (-0.654 - 0.755i)T \)
61 \( 1 + (0.995 - 0.0950i)T \)
71 \( 1 + (-0.888 + 0.458i)T \)
73 \( 1 + (-0.995 + 0.0950i)T \)
79 \( 1 + (0.327 + 0.945i)T \)
83 \( 1 + (0.580 - 0.814i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
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

−32.843049895361991082389289249924, −31.807333348019920142889475359680, −30.69396082828562750692487569978, −29.08402570202743110676650927914, −27.67930396567657459577220535877, −26.68425359846695469343848675031, −26.36392822302508157171971880355, −25.10763855079034886061246268163, −23.65367432177171323038390276894, −22.376590023635770906615249321751, −22.119714462490801095149372034994, −19.894765920910170106989730107180, −19.21877079195790663560318039087, −17.46325254550229095499590335717, −16.60982246934517831021452366821, −15.408451816307563738155821101296, −14.66200430994160360107630338381, −13.52674880561739693932485684495, −11.3020863155409392144627707716, −10.02463076962541092955836134416, −9.123956886897535805268150722, −7.42539972206907012366073653375, −6.38174774208723366470157603529, −4.58125460744182501495840348828, −3.40413140127530842370900574193, 0.182067099169218770077179162878, 1.793396757400814258191200927382, 3.3672689823904322947849885081, 5.21609314381678136842506572581, 7.174083105132360629043470285563, 8.69126521867627102298773595704, 9.4122936263081112880217685982, 11.510370882560545135976879138502, 12.37539977735821755507827182933, 13.025015845638366479417315152689, 14.51934978706317377939586335051, 16.47875578980638668847243351777, 17.596790537324642108645053965938, 18.83767058968774737566047991601, 19.70616667299595254338256379778, 20.375311103543809200757359879127, 22.074455703962986883259130326612, 22.94711788262127004554634182815, 24.46000186949901481000853823899, 25.22720196263293114390647521192, 26.80183719069829829439603511857, 27.94871634095372866626158018997, 28.98106917782230980248264300992, 29.55208264343025137197584251854, 31.0841039589603772478608382315

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