Properties

Label 1-293-293.22-r0-0-0
Degree $1$
Conductor $293$
Sign $-0.891 + 0.452i$
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  + (−0.798 + 0.601i)2-s + (−0.683 + 0.729i)3-s + (0.276 − 0.961i)4-s + (−0.150 − 0.988i)5-s + (0.107 − 0.994i)6-s + (−0.744 − 0.668i)7-s + (0.357 + 0.933i)8-s + (−0.0645 − 0.997i)9-s + (0.714 + 0.699i)10-s + (−0.397 − 0.917i)11-s + (0.512 + 0.858i)12-s + (−0.976 − 0.213i)13-s + (0.996 + 0.0859i)14-s + (0.823 + 0.566i)15-s + (−0.847 − 0.530i)16-s + (−0.798 + 0.601i)17-s + ⋯
L(s)  = 1  + (−0.798 + 0.601i)2-s + (−0.683 + 0.729i)3-s + (0.276 − 0.961i)4-s + (−0.150 − 0.988i)5-s + (0.107 − 0.994i)6-s + (−0.744 − 0.668i)7-s + (0.357 + 0.933i)8-s + (−0.0645 − 0.997i)9-s + (0.714 + 0.699i)10-s + (−0.397 − 0.917i)11-s + (0.512 + 0.858i)12-s + (−0.976 − 0.213i)13-s + (0.996 + 0.0859i)14-s + (0.823 + 0.566i)15-s + (−0.847 − 0.530i)16-s + (−0.798 + 0.601i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(293\)
Sign: $-0.891 + 0.452i$
Analytic conductor: \(1.36068\)
Root analytic conductor: \(1.36068\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{293} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 293,\ (0:\ ),\ -0.891 + 0.452i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03601104790 + 0.1507011351i\)
\(L(\frac12)\) \(\approx\) \(0.03601104790 + 0.1507011351i\)
\(L(1)\) \(\approx\) \(0.3969029835 + 0.09661642126i\)
\(L(1)\) \(\approx\) \(0.3969029835 + 0.09661642126i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad293 \( 1 \)
good2 \( 1 + (-0.798 + 0.601i)T \)
3 \( 1 + (-0.683 + 0.729i)T \)
5 \( 1 + (-0.150 - 0.988i)T \)
7 \( 1 + (-0.744 - 0.668i)T \)
11 \( 1 + (-0.397 - 0.917i)T \)
13 \( 1 + (-0.976 - 0.213i)T \)
17 \( 1 + (-0.798 + 0.601i)T \)
19 \( 1 + (0.823 + 0.566i)T \)
23 \( 1 + (0.107 + 0.994i)T \)
29 \( 1 + (0.0215 + 0.999i)T \)
31 \( 1 + (-0.150 + 0.988i)T \)
37 \( 1 + (0.436 + 0.899i)T \)
41 \( 1 + (0.584 + 0.811i)T \)
43 \( 1 + (-0.890 - 0.455i)T \)
47 \( 1 + (0.869 - 0.493i)T \)
53 \( 1 + (-0.397 - 0.917i)T \)
59 \( 1 + (-0.474 - 0.880i)T \)
61 \( 1 + (-0.925 + 0.377i)T \)
67 \( 1 + (-0.548 - 0.835i)T \)
71 \( 1 + (0.941 - 0.337i)T \)
73 \( 1 + (0.276 + 0.961i)T \)
79 \( 1 + (-0.925 + 0.377i)T \)
83 \( 1 + (0.772 + 0.635i)T \)
89 \( 1 + (-0.925 - 0.377i)T \)
97 \( 1 + (0.985 + 0.171i)T \)
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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

−25.14233225488919339834408401404, −24.41712048953136180338969326686, −22.84831554769053697766336065128, −22.43028358144513222324571477318, −21.69677141950258328627531345763, −20.20263574806432380021742338674, −19.38679277035024789577578012682, −18.608777565646446200425324112726, −18.04555742919685334332932077574, −17.2280808819908944486634949815, −16.07891687163094612515896121675, −15.19894171073380021957597086069, −13.64358907125705709573680384797, −12.61556078865824825517288645681, −11.92386748495212737394862594484, −11.08470508997690034727210275348, −10.07812036491916789521132416571, −9.22971961296209826681803535473, −7.617741832479012362094879795637, −7.124411833261797411256293291605, −6.12791413289488137156141477425, −4.50818461931110105766813700250, −2.65330871239211408018916407873, −2.29938659689415290250579467226, −0.15950470755051451943742691249, 1.16794220520865109818238027127, 3.40586193919832512750640577035, 4.81376796800239455441035961703, 5.5754677719135608065157776552, 6.652635720323283711292057133629, 7.84471504629138582188611283982, 8.9718518579348554807521371812, 9.76962637510826417420792560765, 10.57682990591672274923404295675, 11.60528152794667729857530186059, 12.80678793520058580552069457681, 13.9864163726237931929939753421, 15.33624461197506788197767326073, 16.029679736636741101075772879755, 16.71413332589381950757976838093, 17.250853029066366282440609389104, 18.332041294687230828323786147529, 19.725793187006083107255515686, 20.06477093953590987111993440867, 21.32846591742902134156515650251, 22.330408723872800486299979415964, 23.4998599251690845288306501519, 23.92573266197059108692165455965, 24.949020157024509230505955710662, 26.04758634371755185125216613910

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