Properties

Label 1-293-293.16-r0-0-0
Degree $1$
Conductor $293$
Sign $0.255 + 0.966i$
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  + (0.996 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(293\)
Sign: $0.255 + 0.966i$
Analytic conductor: \(1.36068\)
Root analytic conductor: \(1.36068\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{293} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 293,\ (0:\ ),\ 0.255 + 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.046516329 + 1.576158490i\)
\(L(\frac12)\) \(\approx\) \(2.046516329 + 1.576158490i\)
\(L(1)\) \(\approx\) \(1.887049313 + 0.8440191746i\)
\(L(1)\) \(\approx\) \(1.887049313 + 0.8440191746i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad293 \( 1 \)
good2 \( 1 + (0.996 + 0.0859i)T \)
3 \( 1 + (0.584 + 0.811i)T \)
5 \( 1 + (-0.683 + 0.729i)T \)
7 \( 1 + (0.869 - 0.493i)T \)
11 \( 1 + (-0.890 - 0.455i)T \)
13 \( 1 + (-0.474 + 0.880i)T \)
17 \( 1 + (0.996 + 0.0859i)T \)
19 \( 1 + (-0.991 - 0.128i)T \)
23 \( 1 + (0.512 - 0.858i)T \)
29 \( 1 + (0.107 - 0.994i)T \)
31 \( 1 + (-0.683 - 0.729i)T \)
37 \( 1 + (0.772 + 0.635i)T \)
41 \( 1 + (0.0215 + 0.999i)T \)
43 \( 1 + (0.714 + 0.699i)T \)
47 \( 1 + (-0.847 + 0.530i)T \)
53 \( 1 + (-0.890 - 0.455i)T \)
59 \( 1 + (-0.618 - 0.785i)T \)
61 \( 1 + (0.357 - 0.933i)T \)
67 \( 1 + (-0.234 - 0.972i)T \)
71 \( 1 + (-0.150 + 0.988i)T \)
73 \( 1 + (0.985 - 0.171i)T \)
79 \( 1 + (0.357 - 0.933i)T \)
83 \( 1 + (-0.954 + 0.296i)T \)
89 \( 1 + (0.357 + 0.933i)T \)
97 \( 1 + (0.651 - 0.758i)T \)
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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

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

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