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Properties

Label 1-667-667.534-r0-0-0
Degree $1$
Conductor $667$
Sign $0.702 - 0.711i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $0.702 - 0.711i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (534, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ 0.702 - 0.711i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.114087377 - 0.4657393425i\)
\(L(\frac12)\) \(\approx\) \(1.114087377 - 0.4657393425i\)
\(L(1)\) \(\approx\) \(0.9903112962 + 0.008690875694i\)
\(L(1)\) \(\approx\) \(0.9903112962 + 0.008690875694i\)

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

−22.589320622409198857922189737656, −21.928240422510163809108464511513, −20.91417838033637271089706405103, −20.28647201549797471157671969671, −19.87010366068007559795011503135, −18.65445953334560433890378874129, −18.435366591421444105807710003068, −17.51503967600000424875358057840, −16.02801794646934940671669423815, −15.460618133013253318237375920571, −14.72371311366357193543692469137, −13.58819209268820554440178845010, −12.70071108027428822843715824088, −11.98595960476366305960973224593, −10.97362286798328176766419013865, −10.30222325567867677333479618457, −9.212023246640317948751565497570, −8.39837760195214796263792858949, −7.893748187175311641525206165304, −7.05936125476591835415699664885, −5.13277206950818913805403619527, −4.16253543588035120139748003121, −3.29695878877662779304533066634, −2.443614401467427197618948423655, −1.46033014886591656611354032619, 0.67323407430103334699519306812, 1.82556395280777123237296008555, 3.45196954339993945152735707328, 4.313350696458225872772450766160, 5.220040062786343499495283130442, 6.80418604113861164797546747433, 7.344272220988379809153225077046, 8.262871624415466715385274429506, 8.67793204445631387096931518863, 9.641744830144150611394602459858, 10.87986093812374640709099885383, 11.47860726508339884290537645210, 13.149847530007280012374913639741, 13.6889295740727019263484699071, 14.47090315459290516533659822221, 15.34394712588480718937441724150, 16.09102065350403280183937197611, 16.60482120195856162473149292463, 18.01284365140667520382152715805, 18.472165407980389065296758589440, 19.43502389078331230166458657580, 20.04407874231517654034106531303, 20.70294015236642828092463978844, 21.82718969577717817944853908365, 23.18925386291448121322305799901

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