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Properties

Label 1-67-67.19-r0-0-0
Degree $1$
Conductor $67$
Sign $0.428 + 0.903i$
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  + (0.580 + 0.814i)2-s + (0.841 − 0.540i)3-s + (−0.327 + 0.945i)4-s + (−0.142 + 0.989i)5-s + (0.928 + 0.371i)6-s + (−0.995 + 0.0950i)7-s + (−0.959 + 0.281i)8-s + (0.415 − 0.909i)9-s + (−0.888 + 0.458i)10-s + (0.928 − 0.371i)11-s + (0.235 + 0.971i)12-s + (0.723 − 0.690i)13-s + (−0.654 − 0.755i)14-s + (0.415 + 0.909i)15-s + (−0.786 − 0.618i)16-s + (−0.327 − 0.945i)17-s + ⋯
L(s)  = 1  + (0.580 + 0.814i)2-s + (0.841 − 0.540i)3-s + (−0.327 + 0.945i)4-s + (−0.142 + 0.989i)5-s + (0.928 + 0.371i)6-s + (−0.995 + 0.0950i)7-s + (−0.959 + 0.281i)8-s + (0.415 − 0.909i)9-s + (−0.888 + 0.458i)10-s + (0.928 − 0.371i)11-s + (0.235 + 0.971i)12-s + (0.723 − 0.690i)13-s + (−0.654 − 0.755i)14-s + (0.415 + 0.909i)15-s + (−0.786 − 0.618i)16-s + (−0.327 − 0.945i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.428 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.428 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $0.428 + 0.903i$
Analytic conductor: \(0.311146\)
Root analytic conductor: \(0.311146\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (0:\ ),\ 0.428 + 0.903i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.151895642 + 0.7286209994i\)
\(L(\frac12)\) \(\approx\) \(1.151895642 + 0.7286209994i\)
\(L(1)\) \(\approx\) \(1.325808075 + 0.5852887190i\)
\(L(1)\) \(\approx\) \(1.325808075 + 0.5852887190i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (0.580 + 0.814i)T \)
3 \( 1 + (0.841 - 0.540i)T \)
5 \( 1 + (-0.142 + 0.989i)T \)
7 \( 1 + (-0.995 + 0.0950i)T \)
11 \( 1 + (0.928 - 0.371i)T \)
13 \( 1 + (0.723 - 0.690i)T \)
17 \( 1 + (-0.327 - 0.945i)T \)
19 \( 1 + (-0.995 - 0.0950i)T \)
23 \( 1 + (0.0475 + 0.998i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.723 + 0.690i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.981 + 0.189i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 + (-0.888 - 0.458i)T \)
53 \( 1 + (-0.654 - 0.755i)T \)
59 \( 1 + (-0.959 + 0.281i)T \)
61 \( 1 + (0.928 + 0.371i)T \)
71 \( 1 + (-0.327 + 0.945i)T \)
73 \( 1 + (0.928 + 0.371i)T \)
79 \( 1 + (0.235 + 0.971i)T \)
83 \( 1 + (-0.786 - 0.618i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
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

−31.82119875874085501681468377715, −30.92972103410025565663088918021, −29.80128402506149096479905096860, −28.37043675162607939942177949036, −27.86417838956035540004362952120, −26.3889985787522618486535995617, −25.16693322866659585913241844441, −23.99951479272450390308523802390, −22.69866242186581918837112786031, −21.578543067434913357392105872204, −20.61611020612276901515041259092, −19.71394036482604669226427114228, −19.015474559257508000151023065269, −16.824968304243952430272824028224, −15.656807418871561417804659149680, −14.449492826955760116936414733316, −13.22661363577517724871181174185, −12.41915354494324757094127058498, −10.73850472945389538782190748367, −9.42423513187672925867861482675, −8.705735020742101245207551140696, −6.320323069689670211324457502543, −4.440869617154115172419687136553, −3.702014817079654506743117035635, −1.86259008714808602699632621584, 2.88218102327291466895593629241, 3.7606842651058092817058494103, 6.21309136143613036279300858090, 6.90549699235822913848789107436, 8.25240683973408729672953702391, 9.55258622149860642800255258323, 11.63659681303329397460476364564, 13.08004940392434156726374588071, 13.8785585782713694593975894205, 15.01910091009009023803365652743, 15.87615340260151740753296195426, 17.55217829811927422973181957350, 18.71691097439413519484145437559, 19.72740363729767486322616270289, 21.30301389806114364988212674453, 22.55393943139348972396810265518, 23.27300628657781820096223802696, 24.75470850326891840910238107622, 25.514932070269997562634339152008, 26.28045849027887637424143780845, 27.359711312334742972923511191, 29.65001385565267542943205636424, 30.114418979354375494362368963659, 31.29289218744238831417892275298, 32.1470115977248467011595320220

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