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Properties

Label 1-667-667.244-r0-0-0
Degree $1$
Conductor $667$
Sign $0.387 + 0.921i$
Analytic cond. $3.09753$
Root an. cond. $3.09753$
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  + (0.540 + 0.841i)2-s + (−0.989 − 0.142i)3-s + (−0.415 + 0.909i)4-s + (−0.959 + 0.281i)5-s + (−0.415 − 0.909i)6-s + (0.654 + 0.755i)7-s + (−0.989 + 0.142i)8-s + (0.959 + 0.281i)9-s + (−0.755 − 0.654i)10-s + (0.540 − 0.841i)11-s + (0.540 − 0.841i)12-s + (0.654 − 0.755i)13-s + (−0.281 + 0.959i)14-s + (0.989 − 0.142i)15-s + (−0.654 − 0.755i)16-s + (0.909 − 0.415i)17-s + ⋯
L(s)  = 1  + (0.540 + 0.841i)2-s + (−0.989 − 0.142i)3-s + (−0.415 + 0.909i)4-s + (−0.959 + 0.281i)5-s + (−0.415 − 0.909i)6-s + (0.654 + 0.755i)7-s + (−0.989 + 0.142i)8-s + (0.959 + 0.281i)9-s + (−0.755 − 0.654i)10-s + (0.540 − 0.841i)11-s + (0.540 − 0.841i)12-s + (0.654 − 0.755i)13-s + (−0.281 + 0.959i)14-s + (0.989 − 0.142i)15-s + (−0.654 − 0.755i)16-s + (0.909 − 0.415i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $0.387 + 0.921i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ 0.387 + 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9874912226 + 0.6557859761i\)
\(L(\frac12)\) \(\approx\) \(0.9874912226 + 0.6557859761i\)
\(L(1)\) \(\approx\) \(0.8670268818 + 0.4460610605i\)
\(L(1)\) \(\approx\) \(0.8670268818 + 0.4460610605i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.540 + 0.841i)T \)
3 \( 1 + (-0.989 - 0.142i)T \)
5 \( 1 + (-0.959 + 0.281i)T \)
7 \( 1 + (0.654 + 0.755i)T \)
11 \( 1 + (0.540 - 0.841i)T \)
13 \( 1 + (0.654 - 0.755i)T \)
17 \( 1 + (0.909 - 0.415i)T \)
19 \( 1 + (-0.909 - 0.415i)T \)
31 \( 1 + (0.989 - 0.142i)T \)
37 \( 1 + (0.281 - 0.959i)T \)
41 \( 1 + (0.281 + 0.959i)T \)
43 \( 1 + (0.989 + 0.142i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (-0.654 + 0.755i)T \)
61 \( 1 + (-0.989 + 0.142i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (-0.909 - 0.415i)T \)
79 \( 1 + (0.755 + 0.654i)T \)
83 \( 1 + (0.959 + 0.281i)T \)
89 \( 1 + (0.989 + 0.142i)T \)
97 \( 1 + (-0.281 - 0.959i)T \)
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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

−22.92125213950721636687344673915, −21.806028139018191261378007280085, −20.86995733002187354258518825149, −20.539086526066851919495374380210, −19.29964254193740741189089901734, −18.8336712647045860242236267909, −17.657134269527070435677082400525, −16.9732118244713654897632705611, −16.02250530369134526997455655580, −15.051707899029861483605052014032, −14.34425234156832920069027139355, −13.20449543747382397611933839532, −12.22134264962962781216527781058, −11.871135686707025780580828918389, −10.94502942817690855281195110180, −10.394192361720747232274206284281, −9.32397776526897394988609483364, −8.10293717750813772267609514960, −6.95886499485165875907883529923, −6.04700614078203715582252716839, −4.77606938167001354843764293241, −4.291723213540841997853498276872, −3.630607956260152056208900966343, −1.727147424334204474710753540468, −0.94107756046916884738799255226, 0.8179015370499463259156120743, 2.7957121415416708490945797079, 3.91662512501715612370159411037, 4.77056960829994790185233280674, 5.76133342659888714674395548963, 6.31836412289443709392365390771, 7.46498810170850989661994930903, 8.13579591704737747773240646226, 9.01565805659904504100450115587, 10.64462606363123935888859722211, 11.45526327979191001752822001653, 12.03711579520399663390602055192, 12.80902290092170260191848516471, 13.8927113394234212671530490247, 14.90408963461314386193569532776, 15.51486667893508374926387979091, 16.270350786394127495465924047188, 16.98765682043925569770517194334, 17.98606717357179926199789216082, 18.52735645003889757311343292440, 19.42393751916764827540307673906, 20.926315516809069969088069760061, 21.57547847807481725781811114498, 22.34174422269395157751477762622, 23.1771838184075260876878290859

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