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Properties

Label 1-293-293.95-r0-0-0
Degree $1$
Conductor $293$
Sign $0.933 - 0.358i$
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  + (0.985 − 0.171i)2-s + (−0.317 − 0.948i)3-s + (0.941 − 0.337i)4-s + (−0.0645 + 0.997i)5-s + (−0.474 − 0.880i)6-s + (0.512 + 0.858i)7-s + (0.869 − 0.493i)8-s + (−0.798 + 0.601i)9-s + (0.107 + 0.994i)10-s + (0.584 − 0.811i)11-s + (−0.618 − 0.785i)12-s + (−0.548 + 0.835i)13-s + (0.651 + 0.758i)14-s + (0.966 − 0.255i)15-s + (0.772 − 0.635i)16-s + (0.985 − 0.171i)17-s + ⋯
L(s)  = 1  + (0.985 − 0.171i)2-s + (−0.317 − 0.948i)3-s + (0.941 − 0.337i)4-s + (−0.0645 + 0.997i)5-s + (−0.474 − 0.880i)6-s + (0.512 + 0.858i)7-s + (0.869 − 0.493i)8-s + (−0.798 + 0.601i)9-s + (0.107 + 0.994i)10-s + (0.584 − 0.811i)11-s + (−0.618 − 0.785i)12-s + (−0.548 + 0.835i)13-s + (0.651 + 0.758i)14-s + (0.966 − 0.255i)15-s + (0.772 − 0.635i)16-s + (0.985 − 0.171i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(293\)
Sign: $0.933 - 0.358i$
Analytic conductor: \(1.36068\)
Root analytic conductor: \(1.36068\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{293} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 293,\ (0:\ ),\ 0.933 - 0.358i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.164018222 - 0.4012163539i\)
\(L(\frac12)\) \(\approx\) \(2.164018222 - 0.4012163539i\)
\(L(1)\) \(\approx\) \(1.761740668 - 0.2981582378i\)
\(L(1)\) \(\approx\) \(1.761740668 - 0.2981582378i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad293 \( 1 \)
good2 \( 1 + (0.985 - 0.171i)T \)
3 \( 1 + (-0.317 - 0.948i)T \)
5 \( 1 + (-0.0645 + 0.997i)T \)
7 \( 1 + (0.512 + 0.858i)T \)
11 \( 1 + (0.584 - 0.811i)T \)
13 \( 1 + (-0.548 + 0.835i)T \)
17 \( 1 + (0.985 - 0.171i)T \)
19 \( 1 + (0.966 - 0.255i)T \)
23 \( 1 + (-0.474 + 0.880i)T \)
29 \( 1 + (-0.976 + 0.213i)T \)
31 \( 1 + (-0.0645 - 0.997i)T \)
37 \( 1 + (0.192 - 0.981i)T \)
41 \( 1 + (-0.999 - 0.0430i)T \)
43 \( 1 + (0.0215 - 0.999i)T \)
47 \( 1 + (0.436 + 0.899i)T \)
53 \( 1 + (0.584 - 0.811i)T \)
59 \( 1 + (-0.234 - 0.972i)T \)
61 \( 1 + (-0.744 + 0.668i)T \)
67 \( 1 + (-0.890 - 0.455i)T \)
71 \( 1 + (-0.954 + 0.296i)T \)
73 \( 1 + (0.941 + 0.337i)T \)
79 \( 1 + (-0.744 + 0.668i)T \)
83 \( 1 + (0.823 + 0.566i)T \)
89 \( 1 + (-0.744 - 0.668i)T \)
97 \( 1 + (-0.150 + 0.988i)T \)
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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

−25.266510663812902496520396922483, −24.4988446306114977700213663155, −23.50453543277476197755705243713, −22.84102736152469507806126696778, −22.00033934927075645432779374836, −20.96043700082580294328876000932, −20.28606392502621307004305996915, −19.962967575843409465342482292466, −17.75183470391713375256867483244, −16.80886455205093539687565143211, −16.497109636824287672090129408287, −15.248947953766678115478082724331, −14.58977121879324006891648756380, −13.59122921585751306192534997783, −12.31006021476399656129241562525, −11.84402114042552262728604374249, −10.52355037298285881763386086627, −9.7607896853809562332664749824, −8.26109275762783736209443543196, −7.29382622999248024183125594299, −5.801945661633594790210425127011, −4.93841529395549031488962423447, −4.291644960626392208836891879761, −3.27276239960221684579979573936, −1.38535660030541551316855420581, 1.60840144660537568343726221139, 2.57849012564022855272551972549, 3.661443752191777940093680763278, 5.37471148802164450830936002717, 5.966484763590280368660917831198, 7.07132061873964478128507639426, 7.79748720528283355382214521068, 9.46825709048954289593105347541, 10.99365295201519583870866835063, 11.68531142251350213761364354336, 12.10685653139491849919533035403, 13.552971682196874675121799817499, 14.20227033313690810708154606383, 14.884950304472417078766640478539, 16.124507833053887707618646537777, 17.19667353953530720336646058750, 18.502059696144066146145204889012, 18.95452011330104396773787182299, 19.84959318950129719994450275135, 21.23433952397549663756247748460, 22.092012219654568971748759178464, 22.51188194406859931143922059380, 23.765143770706327983904211598863, 24.21400368135673819542752806414, 25.12371099745856670665418767579

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