Properties

Degree 1
Conductor 293
Sign $0.255 - 0.966i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.996 − 0.0859i)2-s + (0.584 − 0.811i)3-s + (0.985 − 0.171i)4-s + (−0.683 − 0.729i)5-s + (0.512 − 0.858i)6-s + (0.869 + 0.493i)7-s + (0.966 − 0.255i)8-s + (−0.317 − 0.948i)9-s + (−0.744 − 0.668i)10-s + (−0.890 + 0.455i)11-s + (0.436 − 0.899i)12-s + (−0.474 − 0.880i)13-s + (0.908 + 0.417i)14-s + (−0.991 + 0.128i)15-s + (0.941 − 0.337i)16-s + (0.996 − 0.0859i)17-s + ⋯
L(s,χ)  = 1  + (0.996 − 0.0859i)2-s + (0.584 − 0.811i)3-s + (0.985 − 0.171i)4-s + (−0.683 − 0.729i)5-s + (0.512 − 0.858i)6-s + (0.869 + 0.493i)7-s + (0.966 − 0.255i)8-s + (−0.317 − 0.948i)9-s + (−0.744 − 0.668i)10-s + (−0.890 + 0.455i)11-s + (0.436 − 0.899i)12-s + (−0.474 − 0.880i)13-s + (0.908 + 0.417i)14-s + (−0.991 + 0.128i)15-s + (0.941 − 0.337i)16-s + (0.996 − 0.0859i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.255 - 0.966i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.255 - 0.966i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(293\)
\( \varepsilon \)  =  $0.255 - 0.966i$
motivic weight  =  \(0\)
character  :  $\chi_{293} (55, \cdot )$
Sato-Tate  :  $\mu(73)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 293,\ (0:\ ),\ 0.255 - 0.966i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.046516329 - 1.576158490i$
$L(\frac12,\chi)$  $\approx$  $2.046516329 - 1.576158490i$
$L(\chi,1)$  $\approx$  1.887049313 - 0.8440191746i
$L(1,\chi)$  $\approx$  1.887049313 - 0.8440191746i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−25.79072635427855830412862708278, −24.54899278589370609883028814805, −23.66521527428193786312641888129, −23.02292227298479570176787588683, −21.89431890501736238556430461778, −21.206895901193920223996735841194, −20.58876418249612379386131707691, −19.486460206638172622260795002358, −18.73486184871992276359542278768, −16.96710419825761529408166349603, −16.27092832327864974744770098936, −15.18366884461335411033186636715, −14.614467257905438632252448370291, −14.009861067446298155119638036709, −12.826015080536339365723398091842, −11.37141017477130772026571804923, −10.97186132791702725284497530556, −9.94389556297999140902071283628, −8.12068266666796120634654188535, −7.70317073175513061801524664624, −6.3339712913735405900312259774, −4.85923679961303344497292650463, −4.25658973977933519699228220693, −3.16595034072283208344772527433, −2.21125504947857354637513816376, 1.34334002557286380325700190028, 2.480694892536270377072754997485, 3.59685068708239118705639992728, 4.9562195864078427425375820088, 5.644739566237902457476441983650, 7.37294606834497320816063750866, 7.77882488487196709738350478666, 8.88140160568166385614107236190, 10.52735567557791790361064477269, 11.695596303919908049507924632421, 12.59820147873086486438993598144, 12.86219112619087209071636228662, 14.24096689770517149068420219863, 14.99263891829271811817191579066, 15.65644981548331155707331559456, 16.99495597842737740490170600391, 18.11307682626019618580913308232, 19.2185504598945308816248399911, 19.99682441601270144057509630312, 20.806141999071880001576385416147, 21.385376518950134270106537479637, 22.92052218201899407767030463419, 23.672015523693490224432172911314, 24.09010434572816673409426651241, 25.239958085738290218235466641615

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