↑

Properties

Label 1-67-67.47-r0-0-0
Degree $1$
Conductor $67$
Sign $-0.197 + 0.980i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0475 − 0.998i)2-s + (−0.959 − 0.281i)3-s + (−0.995 − 0.0950i)4-s + (−0.654 + 0.755i)5-s + (−0.327 + 0.945i)6-s + (−0.888 + 0.458i)7-s + (−0.142 + 0.989i)8-s + (0.841 + 0.540i)9-s + (0.723 + 0.690i)10-s + (−0.327 − 0.945i)11-s + (0.928 + 0.371i)12-s + (−0.786 + 0.618i)13-s + (0.415 + 0.909i)14-s + (0.841 − 0.540i)15-s + (0.981 + 0.189i)16-s + (−0.995 + 0.0950i)17-s + ⋯
L(s)  = 1  + (0.0475 − 0.998i)2-s + (−0.959 − 0.281i)3-s + (−0.995 − 0.0950i)4-s + (−0.654 + 0.755i)5-s + (−0.327 + 0.945i)6-s + (−0.888 + 0.458i)7-s + (−0.142 + 0.989i)8-s + (0.841 + 0.540i)9-s + (0.723 + 0.690i)10-s + (−0.327 − 0.945i)11-s + (0.928 + 0.371i)12-s + (−0.786 + 0.618i)13-s + (0.415 + 0.909i)14-s + (0.841 − 0.540i)15-s + (0.981 + 0.189i)16-s + (−0.995 + 0.0950i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $-0.197 + 0.980i$
Analytic conductor: \(0.311146\)
Root analytic conductor: \(0.311146\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (0:\ ),\ -0.197 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05618595950 + 0.06864906798i\)
\(L(\frac12)\) \(\approx\) \(0.05618595950 + 0.06864906798i\)
\(L(1)\) \(\approx\) \(0.3923753536 - 0.1316047630i\)
\(L(1)\) \(\approx\) \(0.3923753536 - 0.1316047630i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (0.0475 - 0.998i)T \)
3 \( 1 + (-0.959 - 0.281i)T \)
5 \( 1 + (-0.654 + 0.755i)T \)
7 \( 1 + (-0.888 + 0.458i)T \)
11 \( 1 + (-0.327 - 0.945i)T \)
13 \( 1 + (-0.786 + 0.618i)T \)
17 \( 1 + (-0.995 + 0.0950i)T \)
19 \( 1 + (-0.888 - 0.458i)T \)
23 \( 1 + (0.235 + 0.971i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.786 - 0.618i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.580 + 0.814i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 + (0.723 - 0.690i)T \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 + (-0.142 + 0.989i)T \)
61 \( 1 + (-0.327 + 0.945i)T \)
71 \( 1 + (-0.995 - 0.0950i)T \)
73 \( 1 + (-0.327 + 0.945i)T \)
79 \( 1 + (0.928 + 0.371i)T \)
83 \( 1 + (0.981 + 0.189i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−32.17761980812364529640213423431, −30.92131192428871732228948820341, −29.25210618287967642337390358149, −28.199725295223316247834785647794, −27.29763099417087431515247726154, −26.321214689353830227678358203866, −24.907275702513108303694269371776, −23.848924906983948258735127364896, −22.94091937733733161977259406931, −22.308780318088300859109248855383, −20.59233721616110442601834993277, −19.15601895832063885097273091017, −17.63364995149324236108269204016, −16.83416031815103158730644802354, −15.89019855723712885914557422803, −15.04279340679595266233374625180, −12.955646421856408059347717952944, −12.43552273906192088130685116984, −10.445570389406264058631794606677, −9.263023345965562128824476895317, −7.60169112367552433859732490362, −6.489515327221461885516030418884, −4.99589604037800523111091087422, −4.08971125639467243705956916849, −0.113026981827491454983512948, 2.466481469995473732129560353184, 4.03238904302168898886915905206, 5.70664560595511688421455568404, 7.14591696547580953859032775296, 9.04873664415712548211056397588, 10.58430840176832653816267626907, 11.35394031976430709391557777980, 12.42340374136947189068717303464, 13.502560780393411762671497874125, 15.222798135340272792612992910032, 16.61738028889769678281351588951, 18.02762903418085369578779665730, 19.02056744209320377943315823042, 19.57275649168674657189939352430, 21.71587552650171097757806452450, 22.07290682055946778539972554245, 23.23269662696258407153673427707, 24.10578345807903761682949179787, 26.14654730987155464236963444207, 27.13288462849620584513942034788, 28.17141352458185197358867160395, 29.273732141581814851957059464392, 29.74891958494284445681119125445, 31.144766778152500207893643355508, 31.92616075075056535379517062759

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