Properties

Label 1-293-293.100-r0-0-0
Degree $1$
Conductor $293$
Sign $-0.763 + 0.646i$
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  + (0.357 + 0.933i)2-s + (0.772 + 0.635i)3-s + (−0.744 + 0.668i)4-s + (0.436 + 0.899i)5-s + (−0.317 + 0.948i)6-s + (0.584 − 0.811i)7-s + (−0.890 − 0.455i)8-s + (0.192 + 0.981i)9-s + (−0.683 + 0.729i)10-s + (0.941 + 0.337i)11-s + (−0.999 + 0.0430i)12-s + (−0.798 − 0.601i)13-s + (0.966 + 0.255i)14-s + (−0.234 + 0.972i)15-s + (0.107 − 0.994i)16-s + (0.357 + 0.933i)17-s + ⋯
L(s)  = 1  + (0.357 + 0.933i)2-s + (0.772 + 0.635i)3-s + (−0.744 + 0.668i)4-s + (0.436 + 0.899i)5-s + (−0.317 + 0.948i)6-s + (0.584 − 0.811i)7-s + (−0.890 − 0.455i)8-s + (0.192 + 0.981i)9-s + (−0.683 + 0.729i)10-s + (0.941 + 0.337i)11-s + (−0.999 + 0.0430i)12-s + (−0.798 − 0.601i)13-s + (0.966 + 0.255i)14-s + (−0.234 + 0.972i)15-s + (0.107 − 0.994i)16-s + (0.357 + 0.933i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(293\)
Sign: $-0.763 + 0.646i$
Analytic conductor: \(1.36068\)
Root analytic conductor: \(1.36068\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{293} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 293,\ (0:\ ),\ -0.763 + 0.646i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6776413116 + 1.848329798i\)
\(L(\frac12)\) \(\approx\) \(0.6776413116 + 1.848329798i\)
\(L(1)\) \(\approx\) \(1.084608273 + 1.192051230i\)
\(L(1)\) \(\approx\) \(1.084608273 + 1.192051230i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad293 \( 1 \)
good2 \( 1 + (0.357 + 0.933i)T \)
3 \( 1 + (0.772 + 0.635i)T \)
5 \( 1 + (0.436 + 0.899i)T \)
7 \( 1 + (0.584 - 0.811i)T \)
11 \( 1 + (0.941 + 0.337i)T \)
13 \( 1 + (-0.798 - 0.601i)T \)
17 \( 1 + (0.357 + 0.933i)T \)
19 \( 1 + (-0.234 + 0.972i)T \)
23 \( 1 + (-0.317 - 0.948i)T \)
29 \( 1 + (-0.0645 - 0.997i)T \)
31 \( 1 + (0.436 - 0.899i)T \)
37 \( 1 + (-0.976 - 0.213i)T \)
41 \( 1 + (-0.954 + 0.296i)T \)
43 \( 1 + (-0.150 - 0.988i)T \)
47 \( 1 + (0.0215 - 0.999i)T \)
53 \( 1 + (0.941 + 0.337i)T \)
59 \( 1 + (0.996 + 0.0859i)T \)
61 \( 1 + (-0.397 + 0.917i)T \)
67 \( 1 + (0.985 - 0.171i)T \)
71 \( 1 + (0.512 - 0.858i)T \)
73 \( 1 + (-0.744 - 0.668i)T \)
79 \( 1 + (-0.397 + 0.917i)T \)
83 \( 1 + (-0.474 + 0.880i)T \)
89 \( 1 + (-0.397 - 0.917i)T \)
97 \( 1 + (0.869 + 0.493i)T \)
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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

−24.74890537767390306128176900509, −24.354722208298943749843408313614, −23.48716811089773679074131959159, −21.954868551315680722260479805978, −21.46398786156350924709336050651, −20.54444526264371592221852611357, −19.72917091906983122993073130467, −19.07158713711159472106944877081, −17.99678400217013863872257509298, −17.28729639668543387961271020725, −15.68375840087244701490010179332, −14.4262347544795107201988891938, −14.00157408115078275452556056196, −12.95709120540729987712678883667, −12.04985462674555526159425507045, −11.572451603157518282263306671726, −9.73578988832636507604961892811, −9.07586633508531547054201847432, −8.44669939254189506184858966610, −6.83499923478896947622405519647, −5.46197206852734680101008766067, −4.57038492738756661853803700903, −3.15772327141368107723543999764, −2.044776414025365387562838532018, −1.22668015480783261437760352912, 2.13330091958099989652576585046, 3.59721234905932496481599154354, 4.235610487150590540415735139524, 5.534046255035290222261336048290, 6.75493974463767399048580202489, 7.68829202200046446632671239972, 8.49928956745241866167531968757, 9.91028444900485868408779297960, 10.36322069038506949720907314838, 11.98122890007172535983389682846, 13.34449578619935630803123668764, 14.1834303962409614436442846732, 14.75163603849643634905588000454, 15.274892490414324053821876110336, 16.90364895571109608429894592987, 17.11586101168973289961642387938, 18.42710892434279685165584216550, 19.46988596700806284260529109418, 20.62463128432535241560875129295, 21.440577580586128511919265349435, 22.38218512789357616897438927165, 22.8816586685064616553087119910, 24.26146109889846025387222369592, 25.014239657199063717854352069132, 25.771884614971354527780921035003

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