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Properties

Label 1-67-67.24-r0-0-0
Degree $1$
Conductor $67$
Sign $-0.973 - 0.229i$
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  + (−0.654 − 0.755i)2-s + (0.415 − 0.909i)3-s + (−0.142 + 0.989i)4-s + (−0.959 − 0.281i)5-s + (−0.959 + 0.281i)6-s + (−0.654 − 0.755i)7-s + (0.841 − 0.540i)8-s + (−0.654 − 0.755i)9-s + (0.415 + 0.909i)10-s + (−0.959 − 0.281i)11-s + (0.841 + 0.540i)12-s + (0.841 + 0.540i)13-s + (−0.142 + 0.989i)14-s + (−0.654 + 0.755i)15-s + (−0.959 − 0.281i)16-s + (−0.142 − 0.989i)17-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)2-s + (0.415 − 0.909i)3-s + (−0.142 + 0.989i)4-s + (−0.959 − 0.281i)5-s + (−0.959 + 0.281i)6-s + (−0.654 − 0.755i)7-s + (0.841 − 0.540i)8-s + (−0.654 − 0.755i)9-s + (0.415 + 0.909i)10-s + (−0.959 − 0.281i)11-s + (0.841 + 0.540i)12-s + (0.841 + 0.540i)13-s + (−0.142 + 0.989i)14-s + (−0.654 + 0.755i)15-s + (−0.959 − 0.281i)16-s + (−0.142 − 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $-0.973 - 0.229i$
Analytic conductor: \(0.311146\)
Root analytic conductor: \(0.311146\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (0:\ ),\ -0.973 - 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05830690954 - 0.5003982572i\)
\(L(\frac12)\) \(\approx\) \(0.05830690954 - 0.5003982572i\)
\(L(1)\) \(\approx\) \(0.4245936048 - 0.4677283186i\)
\(L(1)\) \(\approx\) \(0.4245936048 - 0.4677283186i\)

Euler product

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

−32.49142514235839398147036550309, −31.78431114602511451779833950829, −30.70537660915018119051334710808, −28.56209448466940173311010561615, −27.999087712902977165639482906992, −26.91043780665406517391479904093, −25.96251861794503929942731119422, −25.31672168200788542892158588240, −23.62584488781538616136434851254, −22.804012499707876109619947258944, −21.391429050387983725634530658194, −19.836015137470795053120038365092, −19.15789485539074779287855487222, −17.83001194342673725072588740140, −16.20515370428225921768379842245, −15.50921216116048115282332955359, −14.95387921698891928793502730609, −13.17007691970092637423701154346, −11.12823865951529143185453999585, −10.13351785283873489555926133609, −8.752348684790874098276306908056, −7.93190628224990021461759651053, −6.19589378824997518453448965142, −4.68501443599124931819427855334, −2.950337945276335312281759702106, 0.72239044397561115285107441138, 2.7525085086954220305211314830, 4.050941728440657284587688374, 6.804404812061016745161657481151, 7.915580730898157127271983881460, 8.85148922846845010086251600830, 10.528357847662599933095398466179, 11.78994264997716973334982335772, 12.857572116199042802971019120479, 13.77531183749111966791021368398, 15.85764575604328398256380793917, 16.90213849422478187647579753348, 18.516854178142333314741909446, 19.02456036843236463774541006927, 20.188498859206366840598030254150, 20.85161348224520331711583872239, 22.88473137915434452144324335229, 23.5994057133405120120368640029, 25.12503922129107633423797102718, 26.264160033401003801029946637329, 26.99796615924255402903322647139, 28.536193561975804755156265720225, 29.24484489859813223753690056491, 30.36943955271760166321226969293, 31.24452088754522139431371233781

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