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Properties

Label 1-293-293.53-r0-0-0
Degree $1$
Conductor $293$
Sign $0.316 + 0.948i$
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  + (−0.618 − 0.785i)2-s + (0.869 + 0.493i)3-s + (−0.234 + 0.972i)4-s + (−0.744 + 0.668i)5-s + (−0.150 − 0.988i)6-s + (0.651 + 0.758i)7-s + (0.908 − 0.417i)8-s + (0.512 + 0.858i)9-s + (0.985 + 0.171i)10-s + (0.966 + 0.255i)11-s + (−0.683 + 0.729i)12-s + (−0.954 + 0.296i)13-s + (0.192 − 0.981i)14-s + (−0.976 + 0.213i)15-s + (−0.890 − 0.455i)16-s + (−0.618 − 0.785i)17-s + ⋯
L(s)  = 1  + (−0.618 − 0.785i)2-s + (0.869 + 0.493i)3-s + (−0.234 + 0.972i)4-s + (−0.744 + 0.668i)5-s + (−0.150 − 0.988i)6-s + (0.651 + 0.758i)7-s + (0.908 − 0.417i)8-s + (0.512 + 0.858i)9-s + (0.985 + 0.171i)10-s + (0.966 + 0.255i)11-s + (−0.683 + 0.729i)12-s + (−0.954 + 0.296i)13-s + (0.192 − 0.981i)14-s + (−0.976 + 0.213i)15-s + (−0.890 − 0.455i)16-s + (−0.618 − 0.785i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(293\)
Sign: $0.316 + 0.948i$
Analytic conductor: \(1.36068\)
Root analytic conductor: \(1.36068\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{293} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 293,\ (0:\ ),\ 0.316 + 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8246757493 + 0.5943338431i\)
\(L(\frac12)\) \(\approx\) \(0.8246757493 + 0.5943338431i\)
\(L(1)\) \(\approx\) \(0.9124327063 + 0.1841539498i\)
\(L(1)\) \(\approx\) \(0.9124327063 + 0.1841539498i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad293 \( 1 \)
good2 \( 1 + (-0.618 - 0.785i)T \)
3 \( 1 + (0.869 + 0.493i)T \)
5 \( 1 + (-0.744 + 0.668i)T \)
7 \( 1 + (0.651 + 0.758i)T \)
11 \( 1 + (0.966 + 0.255i)T \)
13 \( 1 + (-0.954 + 0.296i)T \)
17 \( 1 + (-0.618 - 0.785i)T \)
19 \( 1 + (-0.976 + 0.213i)T \)
23 \( 1 + (-0.150 + 0.988i)T \)
29 \( 1 + (0.941 + 0.337i)T \)
31 \( 1 + (-0.744 - 0.668i)T \)
37 \( 1 + (0.584 + 0.811i)T \)
41 \( 1 + (-0.847 - 0.530i)T \)
43 \( 1 + (0.276 - 0.961i)T \)
47 \( 1 + (-0.397 + 0.917i)T \)
53 \( 1 + (0.966 + 0.255i)T \)
59 \( 1 + (-0.0645 + 0.997i)T \)
61 \( 1 + (0.996 - 0.0859i)T \)
67 \( 1 + (-0.991 + 0.128i)T \)
71 \( 1 + (0.714 - 0.699i)T \)
73 \( 1 + (-0.234 - 0.972i)T \)
79 \( 1 + (0.996 - 0.0859i)T \)
83 \( 1 + (0.0215 + 0.999i)T \)
89 \( 1 + (0.996 + 0.0859i)T \)
97 \( 1 + (-0.925 - 0.377i)T \)
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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

−25.0102305218622810052376559661, −24.55114646943907279851301027034, −23.833149535340372425453342016368, −23.11489921393115264532412082260, −21.560727177193300070407481432884, −20.14446187189221780348828931196, −19.80615212672818089557911766147, −19.122565576463607501091276399806, −17.81951128262457853607705147314, −17.11122153385784474859932529352, −16.21835041672187804726960145171, −14.8521783831384530242967863900, −14.66006018468707028784356736617, −13.42449451027337835724166856262, −12.41505415311037780052077723901, −11.115325569734835324262742304716, −9.919756069787236666537760685563, −8.66391384270837328052136718101, −8.31796081745970804389403681914, −7.283686415371794071138863516733, −6.48414473408317927282610307662, −4.73702946990192561343768516827, −3.95908761192970370910163270101, −1.973542200368101956886645122057, −0.7664195713823130374570593117, 1.912551505773731648161637567028, 2.73189575635126415722855815808, 3.90856220614720318598590935091, 4.72602344912232845511656375214, 6.94196531359161726704290818786, 7.82941503715973865373563063067, 8.781062596188825222951237338589, 9.51313176451555867515506794208, 10.605860394160063501861091989482, 11.58464416570279614103687631607, 12.231107630913334251599447796917, 13.7071068688659082897860040074, 14.70294584947026957068276323847, 15.371631846910320664995313973578, 16.54223775843375985232646829798, 17.67601787951235901387380214874, 18.677547848599395763088972681364, 19.423542306948096007693967423925, 19.99499789914999681576013276288, 21.01412446282252181922172378882, 22.0002620715660212644041029560, 22.317703580997621704205102156665, 23.95827318370869309611177401178, 25.16277989729376252110486333575, 25.709088625856300747125803584804

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