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Properties

Label 1-293-293.33-r0-0-0
Degree $1$
Conductor $293$
Sign $0.479 + 0.877i$
Analytic cond. $1.36068$
Root an. cond. $1.36068$
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  + (0.908 − 0.417i)2-s + (0.0215 + 0.999i)3-s + (0.651 − 0.758i)4-s + (0.584 + 0.811i)5-s + (0.436 + 0.899i)6-s + (−0.847 + 0.530i)7-s + (0.276 − 0.961i)8-s + (−0.999 + 0.0430i)9-s + (0.869 + 0.493i)10-s + (0.714 + 0.699i)11-s + (0.772 + 0.635i)12-s + (−0.618 + 0.785i)13-s + (−0.548 + 0.835i)14-s + (−0.798 + 0.601i)15-s + (−0.150 − 0.988i)16-s + (0.908 − 0.417i)17-s + ⋯
L(s)  = 1  + (0.908 − 0.417i)2-s + (0.0215 + 0.999i)3-s + (0.651 − 0.758i)4-s + (0.584 + 0.811i)5-s + (0.436 + 0.899i)6-s + (−0.847 + 0.530i)7-s + (0.276 − 0.961i)8-s + (−0.999 + 0.0430i)9-s + (0.869 + 0.493i)10-s + (0.714 + 0.699i)11-s + (0.772 + 0.635i)12-s + (−0.618 + 0.785i)13-s + (−0.548 + 0.835i)14-s + (−0.798 + 0.601i)15-s + (−0.150 − 0.988i)16-s + (0.908 − 0.417i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(293\)
Sign: $0.479 + 0.877i$
Analytic conductor: \(1.36068\)
Root analytic conductor: \(1.36068\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{293} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 293,\ (0:\ ),\ 0.479 + 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.808367040 + 1.072884227i\)
\(L(\frac12)\) \(\approx\) \(1.808367040 + 1.072884227i\)
\(L(1)\) \(\approx\) \(1.673201140 + 0.4638993373i\)
\(L(1)\) \(\approx\) \(1.673201140 + 0.4638993373i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad293 \( 1 \)
good2 \( 1 + (0.908 - 0.417i)T \)
3 \( 1 + (0.0215 + 0.999i)T \)
5 \( 1 + (0.584 + 0.811i)T \)
7 \( 1 + (-0.847 + 0.530i)T \)
11 \( 1 + (0.714 + 0.699i)T \)
13 \( 1 + (-0.618 + 0.785i)T \)
17 \( 1 + (0.908 - 0.417i)T \)
19 \( 1 + (-0.798 + 0.601i)T \)
23 \( 1 + (0.436 - 0.899i)T \)
29 \( 1 + (0.512 + 0.858i)T \)
31 \( 1 + (0.584 - 0.811i)T \)
37 \( 1 + (-0.954 + 0.296i)T \)
41 \( 1 + (0.107 - 0.994i)T \)
43 \( 1 + (-0.744 + 0.668i)T \)
47 \( 1 + (0.941 - 0.337i)T \)
53 \( 1 + (0.714 + 0.699i)T \)
59 \( 1 + (0.192 - 0.981i)T \)
61 \( 1 + (0.966 - 0.255i)T \)
67 \( 1 + (-0.925 + 0.377i)T \)
71 \( 1 + (-0.683 - 0.729i)T \)
73 \( 1 + (0.651 + 0.758i)T \)
79 \( 1 + (0.966 - 0.255i)T \)
83 \( 1 + (-0.0645 - 0.997i)T \)
89 \( 1 + (0.966 + 0.255i)T \)
97 \( 1 + (-0.397 - 0.917i)T \)
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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

−25.25686730602200987250888833586, −24.36274896242384039542248566824, −23.58899153096996314352121133369, −22.83673469608391916902176468748, −21.83717908424714215215679411001, −20.92286306604434677577319873154, −19.73149997225177248750384194763, −19.37021169205724523940898573934, −17.521641898327450595101353833257, −17.116366280652753148958484365231, −16.2858146265230564497207753255, −14.98308975997327225222787632761, −13.84128514429827433711117657219, −13.35457634496326205312429250252, −12.54085151227749141271251958758, −11.8391148444507093049494870298, −10.37662871476366412674018975252, −8.9215012967927332147137390041, −7.95482308906968285664237451623, −6.84583832837417305493246571487, −6.05891584334402228218851913905, −5.186066815293755330058244424943, −3.6835150585234226159523478004, −2.59520601339548373713079876872, −1.091960564559894554149083361514, 2.12032404711081477589515345494, 3.02304573506851328199949443579, 4.0215498860050332133643051704, 5.14741748661659474463738512962, 6.19951335040621854662150568191, 6.95537336865788350459352056928, 9.07093322790647771341815711968, 9.92144982734654028232601155425, 10.45672623459465208817005546399, 11.74205865181857967630090590426, 12.44424121682478175545523789201, 13.85601570171894632820448478916, 14.60865481149298260144215776254, 15.130982312629569571313140502704, 16.29546699988245323748558295510, 17.11403665031849562218482618498, 18.73376164335889794759891233153, 19.352382774191534905150129349068, 20.51878400703917972459964388984, 21.31429554725538403581533041981, 22.09247955057217833061651978201, 22.58509736325665899050893929190, 23.31878037723384115545337654725, 24.948349215615415593648702827770, 25.46971445444889601931610983492

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