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Properties

Label 1-67-67.56-r0-0-0
Degree $1$
Conductor $67$
Sign $0.283 + 0.959i$
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  + (−0.786 + 0.618i)2-s + (−0.654 + 0.755i)3-s + (0.235 − 0.971i)4-s + (0.841 − 0.540i)5-s + (0.0475 − 0.998i)6-s + (0.928 + 0.371i)7-s + (0.415 + 0.909i)8-s + (−0.142 − 0.989i)9-s + (−0.327 + 0.945i)10-s + (0.0475 + 0.998i)11-s + (0.580 + 0.814i)12-s + (−0.995 + 0.0950i)13-s + (−0.959 + 0.281i)14-s + (−0.142 + 0.989i)15-s + (−0.888 − 0.458i)16-s + (0.235 + 0.971i)17-s + ⋯
L(s)  = 1  + (−0.786 + 0.618i)2-s + (−0.654 + 0.755i)3-s + (0.235 − 0.971i)4-s + (0.841 − 0.540i)5-s + (0.0475 − 0.998i)6-s + (0.928 + 0.371i)7-s + (0.415 + 0.909i)8-s + (−0.142 − 0.989i)9-s + (−0.327 + 0.945i)10-s + (0.0475 + 0.998i)11-s + (0.580 + 0.814i)12-s + (−0.995 + 0.0950i)13-s + (−0.959 + 0.281i)14-s + (−0.142 + 0.989i)15-s + (−0.888 − 0.458i)16-s + (0.235 + 0.971i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $0.283 + 0.959i$
Analytic conductor: \(0.311146\)
Root analytic conductor: \(0.311146\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (0:\ ),\ 0.283 + 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5116210080 + 0.3824134433i\)
\(L(\frac12)\) \(\approx\) \(0.5116210080 + 0.3824134433i\)
\(L(1)\) \(\approx\) \(0.6508603162 + 0.3115735324i\)
\(L(1)\) \(\approx\) \(0.6508603162 + 0.3115735324i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (-0.786 + 0.618i)T \)
3 \( 1 + (-0.654 + 0.755i)T \)
5 \( 1 + (0.841 - 0.540i)T \)
7 \( 1 + (0.928 + 0.371i)T \)
11 \( 1 + (0.0475 + 0.998i)T \)
13 \( 1 + (-0.995 + 0.0950i)T \)
17 \( 1 + (0.235 + 0.971i)T \)
19 \( 1 + (0.928 - 0.371i)T \)
23 \( 1 + (0.981 + 0.189i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.995 - 0.0950i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.723 - 0.690i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (-0.327 - 0.945i)T \)
53 \( 1 + (-0.959 + 0.281i)T \)
59 \( 1 + (0.415 + 0.909i)T \)
61 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (0.235 - 0.971i)T \)
73 \( 1 + (0.0475 - 0.998i)T \)
79 \( 1 + (0.580 + 0.814i)T \)
83 \( 1 + (-0.888 - 0.458i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
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

−31.31663363627132706445819317309, −30.19430096766526233820871946449, −29.498275236458091783266251652463, −28.877382097555655682250605230508, −27.377980862003480697580033909321, −26.658370794016340068906460829754, −25.07626100062561817942368830801, −24.39592474753478529480308692338, −22.655424212081156817979167662685, −21.71527705957596499623672500136, −20.571047551921477889068385682429, −19.07215492241585650509054547071, −18.25571070878207072397800913490, −17.37632043420776959462020239647, −16.53784655422826880189149828929, −14.25706445442116852613897594257, −13.17828453499795267148072134784, −11.670530138065214141310213461910, −10.9439128483701088388861991459, −9.6549646307402747732252962982, −7.934430775507692782876974475750, −6.92510580785494756973639825051, −5.24873633831191262004010383208, −2.77644808941050572530356814260, −1.30573367910127874042384673796, 1.73393371002523266968049104033, 4.87707164385907087732548072677, 5.5282736102232391813432103485, 7.18457258664405498497523274124, 8.917658293826800817898006415883, 9.75718964277253090260535305797, 10.916037915155044189430855362213, 12.36852962459091072852452104866, 14.452918374746916554439865098181, 15.24039651595282928092580347824, 16.67998524542815225915458986583, 17.42082952620156083898981271476, 18.13869202660954032422179554343, 20.02028162619582332396245195017, 21.05999613884446391166397862923, 22.19725699254338371440428332518, 23.685334839911422874900522137943, 24.60724951176182595125327626057, 25.72330501456284272233296006818, 26.8836862098245404582012592169, 27.9309997256781868371016524387, 28.50199533061421009425901343546, 29.58008925460400875689777280194, 31.46088487296462683791568544089, 32.888805194309940621231291840990

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