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Properties

Label 1-67-67.41-r1-0-0
Degree $1$
Conductor $67$
Sign $-0.106 + 0.994i$
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  + (0.327 − 0.945i)2-s + (−0.415 + 0.909i)3-s + (−0.786 − 0.618i)4-s + (0.959 + 0.281i)5-s + (0.723 + 0.690i)6-s + (−0.981 + 0.189i)7-s + (−0.841 + 0.540i)8-s + (−0.654 − 0.755i)9-s + (0.580 − 0.814i)10-s + (−0.723 + 0.690i)11-s + (0.888 − 0.458i)12-s + (−0.0475 + 0.998i)13-s + (−0.142 + 0.989i)14-s + (−0.654 + 0.755i)15-s + (0.235 + 0.971i)16-s + (−0.786 + 0.618i)17-s + ⋯
L(s)  = 1  + (0.327 − 0.945i)2-s + (−0.415 + 0.909i)3-s + (−0.786 − 0.618i)4-s + (0.959 + 0.281i)5-s + (0.723 + 0.690i)6-s + (−0.981 + 0.189i)7-s + (−0.841 + 0.540i)8-s + (−0.654 − 0.755i)9-s + (0.580 − 0.814i)10-s + (−0.723 + 0.690i)11-s + (0.888 − 0.458i)12-s + (−0.0475 + 0.998i)13-s + (−0.142 + 0.989i)14-s + (−0.654 + 0.755i)15-s + (0.235 + 0.971i)16-s + (−0.786 + 0.618i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $-0.106 + 0.994i$
Analytic conductor: \(7.20014\)
Root analytic conductor: \(7.20014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (1:\ ),\ -0.106 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5520158873 + 0.6139908458i\)
\(L(\frac12)\) \(\approx\) \(0.5520158873 + 0.6139908458i\)
\(L(1)\) \(\approx\) \(0.8488860164 + 0.04797255198i\)
\(L(1)\) \(\approx\) \(0.8488860164 + 0.04797255198i\)

Euler product

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

−31.72808160275900098073514204444, −30.32126480109557222750390704485, −29.33154791448253861172954275967, −28.44460818343102573804431108200, −26.6801874209954906578962343886, −25.599786883359557671726755287, −24.79108268715034329452781519320, −23.91447544290649462057981010358, −22.6508981939472949703789446965, −21.98929806145809218440543506296, −20.24680849608537424503055593762, −18.55903505739994934254848599940, −17.78150183015349289553925754962, −16.70065649431980465502975138599, −15.70598379615480268719790825699, −13.62119597830557520708193379837, −13.46945521710997153051734392190, −12.208472709077949256236563828464, −10.1759947972467892810307435623, −8.651137961809115579033161739558, −7.266053673836383793023508457203, −6.089658808375137337834455089548, −5.27485905350728444506560137912, −2.89564921111777563111612116434, −0.382606951034480252559950432262, 2.227499873520985536167615585389, 3.725372025844090140932956887005, 5.20493195297284155144700768900, 6.34140025834061638822167177895, 9.232006870181987560401375752611, 9.856153061743104833162651061, 10.878341275710459579010947674, 12.2803286600120292917807315288, 13.48892415840571929168865775508, 14.72603248099214426295995386092, 16.05470088644677218040479797383, 17.52296270489728090260161440356, 18.54535246363206555287275497506, 20.03914349893753228181885216863, 21.0629946092247618273292255607, 22.052772270015003601952248197813, 22.5671659688996099838436083056, 23.93140474384313249396265849222, 25.93131749223455247492561185266, 26.5031480302080004960696941945, 28.18727352009459097339086135382, 28.75113838134154092787587939012, 29.49960555970028332284898259270, 31.0417038756131263023420617980, 32.00341742403314592653030319136

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