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Properties

Label 1-67-67.62-r0-0-0
Degree $1$
Conductor $67$
Sign $0.0678 - 0.997i$
Analytic cond. $0.311146$
Root an. cond. $0.311146$
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  + (−0.142 − 0.989i)2-s + (−0.654 + 0.755i)3-s + (−0.959 + 0.281i)4-s + (0.841 − 0.540i)5-s + (0.841 + 0.540i)6-s + (−0.142 − 0.989i)7-s + (0.415 + 0.909i)8-s + (−0.142 − 0.989i)9-s + (−0.654 − 0.755i)10-s + (0.841 − 0.540i)11-s + (0.415 − 0.909i)12-s + (0.415 − 0.909i)13-s + (−0.959 + 0.281i)14-s + (−0.142 + 0.989i)15-s + (0.841 − 0.540i)16-s + (−0.959 − 0.281i)17-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)2-s + (−0.654 + 0.755i)3-s + (−0.959 + 0.281i)4-s + (0.841 − 0.540i)5-s + (0.841 + 0.540i)6-s + (−0.142 − 0.989i)7-s + (0.415 + 0.909i)8-s + (−0.142 − 0.989i)9-s + (−0.654 − 0.755i)10-s + (0.841 − 0.540i)11-s + (0.415 − 0.909i)12-s + (0.415 − 0.909i)13-s + (−0.959 + 0.281i)14-s + (−0.142 + 0.989i)15-s + (0.841 − 0.540i)16-s + (−0.959 − 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0678 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0678 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $0.0678 - 0.997i$
Analytic conductor: \(0.311146\)
Root analytic conductor: \(0.311146\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (0:\ ),\ 0.0678 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5287150598 - 0.4939567843i\)
\(L(\frac12)\) \(\approx\) \(0.5287150598 - 0.4939567843i\)
\(L(1)\) \(\approx\) \(0.7250076969 - 0.3846948307i\)
\(L(1)\) \(\approx\) \(0.7250076969 - 0.3846948307i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (-0.142 - 0.989i)T \)
3 \( 1 + (-0.654 + 0.755i)T \)
5 \( 1 + (0.841 - 0.540i)T \)
7 \( 1 + (-0.142 - 0.989i)T \)
11 \( 1 + (0.841 - 0.540i)T \)
13 \( 1 + (0.415 - 0.909i)T \)
17 \( 1 + (-0.959 - 0.281i)T \)
19 \( 1 + (-0.142 + 0.989i)T \)
23 \( 1 + (-0.654 + 0.755i)T \)
29 \( 1 + T \)
31 \( 1 + (0.415 + 0.909i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.959 - 0.281i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (-0.654 + 0.755i)T \)
53 \( 1 + (-0.959 + 0.281i)T \)
59 \( 1 + (0.415 + 0.909i)T \)
61 \( 1 + (0.841 + 0.540i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (0.841 + 0.540i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (0.841 - 0.540i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + T \)
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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

−32.66122837653272190138685206967, −31.095478345399793587601800786848, −30.2573048520866359585197508666, −28.632479595607787828151017628892, −28.19188200412800049751010628856, −26.49058798435365789341102460593, −25.398051848999565923274670482640, −24.725583861779717770274176848304, −23.63073716801301144712335503379, −22.28822479312421072110172262409, −21.87391783467585768130984475541, −19.39438961538534068482371114188, −18.33490326790241230400662315053, −17.68462603104157829865659546397, −16.60517632383947675368290834212, −15.18171961414142079390451918784, −13.96891143548023203186432454746, −12.92403902122432943690314847637, −11.44613638764016576751858991950, −9.73699394514464922850335963066, −8.51357026249651207638058596946, −6.56356146027812669108332207971, −6.41739294707983558165438022962, −4.79539015325641686533966236811, −2.00681134059704995185373529770, 1.1470664372553927713297356992, 3.49692364855410980533804181845, 4.71189507750891936266818758080, 6.14220077986886792201507628855, 8.561122386125019388686072587878, 9.81526168598399783790040725009, 10.560447979386750470996791201372, 11.80571138596262074446424675142, 13.16032982271417220036855616126, 14.19808776471706635371952430286, 16.217705285324934570056258694595, 17.24151824426061613921031744920, 17.94179267024243624904041313714, 19.83069032362391753228906230511, 20.614051302128333442401090415063, 21.65061407283893246492470236593, 22.52784943943369997094050246806, 23.617586930593237641714420695516, 25.36564206549339990592749330833, 26.80900489936403573910237369548, 27.43891556981005184519687986113, 28.65140431525968510354482656262, 29.39378906460510106221438071060, 30.22931027850929801347934023355, 32.041072907353938766663169516657

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