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Properties

Label 1-293-293.77-r0-0-0
Degree $1$
Conductor $293$
Sign $-0.913 - 0.406i$
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  + (0.714 + 0.699i)2-s + (−0.618 + 0.785i)3-s + (0.0215 + 0.999i)4-s + (−0.474 + 0.880i)5-s + (−0.991 + 0.128i)6-s + (−0.0645 + 0.997i)7-s + (−0.683 + 0.729i)8-s + (−0.234 − 0.972i)9-s + (−0.954 + 0.296i)10-s + (0.436 + 0.899i)11-s + (−0.798 − 0.601i)12-s + (0.966 − 0.255i)13-s + (−0.744 + 0.668i)14-s + (−0.397 − 0.917i)15-s + (−0.999 + 0.0430i)16-s + (0.714 + 0.699i)17-s + ⋯
L(s)  = 1  + (0.714 + 0.699i)2-s + (−0.618 + 0.785i)3-s + (0.0215 + 0.999i)4-s + (−0.474 + 0.880i)5-s + (−0.991 + 0.128i)6-s + (−0.0645 + 0.997i)7-s + (−0.683 + 0.729i)8-s + (−0.234 − 0.972i)9-s + (−0.954 + 0.296i)10-s + (0.436 + 0.899i)11-s + (−0.798 − 0.601i)12-s + (0.966 − 0.255i)13-s + (−0.744 + 0.668i)14-s + (−0.397 − 0.917i)15-s + (−0.999 + 0.0430i)16-s + (0.714 + 0.699i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(293\)
Sign: $-0.913 - 0.406i$
Analytic conductor: \(1.36068\)
Root analytic conductor: \(1.36068\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{293} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 293,\ (0:\ ),\ -0.913 - 0.406i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2587503807 + 1.219495587i\)
\(L(\frac12)\) \(\approx\) \(-0.2587503807 + 1.219495587i\)
\(L(1)\) \(\approx\) \(0.5691927672 + 0.9878054377i\)
\(L(1)\) \(\approx\) \(0.5691927672 + 0.9878054377i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad293 \( 1 \)
good2 \( 1 + (0.714 + 0.699i)T \)
3 \( 1 + (-0.618 + 0.785i)T \)
5 \( 1 + (-0.474 + 0.880i)T \)
7 \( 1 + (-0.0645 + 0.997i)T \)
11 \( 1 + (0.436 + 0.899i)T \)
13 \( 1 + (0.966 - 0.255i)T \)
17 \( 1 + (0.714 + 0.699i)T \)
19 \( 1 + (-0.397 - 0.917i)T \)
23 \( 1 + (-0.991 - 0.128i)T \)
29 \( 1 + (0.823 - 0.566i)T \)
31 \( 1 + (-0.474 - 0.880i)T \)
37 \( 1 + (0.996 - 0.0859i)T \)
41 \( 1 + (0.192 + 0.981i)T \)
43 \( 1 + (0.772 + 0.635i)T \)
47 \( 1 + (-0.317 - 0.948i)T \)
53 \( 1 + (0.436 + 0.899i)T \)
59 \( 1 + (0.276 - 0.961i)T \)
61 \( 1 + (-0.150 + 0.988i)T \)
67 \( 1 + (-0.847 + 0.530i)T \)
71 \( 1 + (-0.976 + 0.213i)T \)
73 \( 1 + (0.0215 - 0.999i)T \)
79 \( 1 + (-0.150 + 0.988i)T \)
83 \( 1 + (0.908 + 0.417i)T \)
89 \( 1 + (-0.150 - 0.988i)T \)
97 \( 1 + (0.107 - 0.994i)T \)
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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

−24.531521438749361976974762990613, −23.664624962041806827665589002916, −23.39203041772635926565727384042, −22.44620414640653818386809827053, −21.26513310338662959899073690024, −20.42695047950798689213349068220, −19.562292355393143512292793423533, −18.88225270609797694643056819455, −17.794825832311669672775027918829, −16.38004424089961658416362632757, −16.196405245434027184138062159889, −14.2126579364138572176471238343, −13.72427783205116008483893074626, −12.74160801187382907640054363059, −11.97719723569372969746899851071, −11.183889186577193039223519659269, −10.292062138983446668048538597151, −8.81493566582879355672623608744, −7.65470479493479384215004728937, −6.38854666904978662991260783486, −5.53315419356843168089522038690, −4.33712932496185441248961457777, −3.43052775264630394506949591931, −1.54771294727268298449927386107, −0.76419926287297270693996289677, 2.591659602888817597504283191139, 3.74985456030626555326820942243, 4.5536946955921550064182161945, 5.9599451082906051953335413338, 6.343665661254358283308984821221, 7.7307758048731250311726434477, 8.85098938407534420239613596403, 10.0695499289029081140442477066, 11.324250646570714868374612672852, 11.92869344076654017506206906223, 12.92254020751372985202311962949, 14.448182015139081480332434838636, 15.10029018386218100586395134279, 15.62295749672361967439211342601, 16.54117253770928565464571559455, 17.73254586385056338475192829609, 18.2804622897604928734126134469, 19.77261709203947099178461483307, 21.01029360555201372363462681072, 21.802825397766888790736488537145, 22.404712753705699854064117258601, 23.18021638854402526506228399866, 23.809123308087711708111864883253, 25.23841591791352794616729691485, 25.878234921627911325456594473868

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