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Properties

Label 1-67-67.21-r0-0-0
Degree $1$
Conductor $67$
Sign $0.375 - 0.926i$
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  + (0.928 − 0.371i)2-s + (−0.654 − 0.755i)3-s + (0.723 − 0.690i)4-s + (0.841 + 0.540i)5-s + (−0.888 − 0.458i)6-s + (−0.786 − 0.618i)7-s + (0.415 − 0.909i)8-s + (−0.142 + 0.989i)9-s + (0.981 + 0.189i)10-s + (−0.888 + 0.458i)11-s + (−0.995 − 0.0950i)12-s + (0.580 − 0.814i)13-s + (−0.959 − 0.281i)14-s + (−0.142 − 0.989i)15-s + (0.0475 − 0.998i)16-s + (0.723 + 0.690i)17-s + ⋯
L(s)  = 1  + (0.928 − 0.371i)2-s + (−0.654 − 0.755i)3-s + (0.723 − 0.690i)4-s + (0.841 + 0.540i)5-s + (−0.888 − 0.458i)6-s + (−0.786 − 0.618i)7-s + (0.415 − 0.909i)8-s + (−0.142 + 0.989i)9-s + (0.981 + 0.189i)10-s + (−0.888 + 0.458i)11-s + (−0.995 − 0.0950i)12-s + (0.580 − 0.814i)13-s + (−0.959 − 0.281i)14-s + (−0.142 − 0.989i)15-s + (0.0475 − 0.998i)16-s + (0.723 + 0.690i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $0.375 - 0.926i$
Analytic conductor: \(0.311146\)
Root analytic conductor: \(0.311146\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (0:\ ),\ 0.375 - 0.926i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.087471924 - 0.7327900463i\)
\(L(\frac12)\) \(\approx\) \(1.087471924 - 0.7327900463i\)
\(L(1)\) \(\approx\) \(1.278573352 - 0.5743229508i\)
\(L(1)\) \(\approx\) \(1.278573352 - 0.5743229508i\)

Euler product

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

−32.07496792764489544869995548664, −31.81945146147702595308087899872, −29.931041897766896371018190420487, −28.879171742966292204969116992035, −28.29433053535735263942712603864, −26.359663069982704456956639139786, −25.63549385716606884411772041455, −24.31693111697834425098970579251, −23.249296380938129170229151360392, −22.15687481731325534282984266464, −21.31228057569073383440072953517, −20.629642766427835611406098219463, −18.53539898480372781410092783551, −16.95653440106947764977861019027, −16.26150769127054304335301868095, −15.306357251352856859905442335867, −13.76164243887809439643716179585, −12.71142806504819936818704106696, −11.524133423351442670562047677972, −10.05146470744842210278246532263, −8.68251242955646752514935869217, −6.421893308460851825829045982914, −5.65812928599237426362744419308, −4.46018817173724945986664015565, −2.74104820432264326626869312507, 1.74008142357421962617958603502, 3.32608435537251196173089790902, 5.40654085562163062318463529704, 6.27730236175396181505914354024, 7.46000078942592479662127740898, 10.22241885619683500033679469987, 10.72302444928202906014156170370, 12.528372588727994384028748407273, 13.1484353606624199006599749648, 14.19408501656798516534106167964, 15.75842131089727402246037150628, 17.14442705386611811279994822117, 18.40906765312079252365233595641, 19.464696819554506179333223948947, 20.85227354046664528214806199676, 22.024291433569651255976490700760, 23.08045287086308700816134230320, 23.54537905906906897894748025333, 25.17704744715802318762679125389, 25.81740491855966299100874846646, 27.97604151871538320332719173483, 28.96748420168093141409603129704, 29.79890615888678197144788528595, 30.31428210321798099302171455214, 31.745254994657900528182679491012

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