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Properties

Label 1-293-293.69-r0-0-0
Degree $1$
Conductor $293$
Sign $-0.579 + 0.815i$
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  + (0.0215 + 0.999i)2-s + (−0.234 − 0.972i)3-s + (−0.999 + 0.0430i)4-s + (−0.548 − 0.835i)5-s + (0.966 − 0.255i)6-s + (−0.991 − 0.128i)7-s + (−0.0645 − 0.997i)8-s + (−0.890 + 0.455i)9-s + (0.823 − 0.566i)10-s + (−0.618 + 0.785i)11-s + (0.276 + 0.961i)12-s + (0.869 − 0.493i)13-s + (0.107 − 0.994i)14-s + (−0.683 + 0.729i)15-s + (0.996 − 0.0859i)16-s + (0.0215 + 0.999i)17-s + ⋯
L(s)  = 1  + (0.0215 + 0.999i)2-s + (−0.234 − 0.972i)3-s + (−0.999 + 0.0430i)4-s + (−0.548 − 0.835i)5-s + (0.966 − 0.255i)6-s + (−0.991 − 0.128i)7-s + (−0.0645 − 0.997i)8-s + (−0.890 + 0.455i)9-s + (0.823 − 0.566i)10-s + (−0.618 + 0.785i)11-s + (0.276 + 0.961i)12-s + (0.869 − 0.493i)13-s + (0.107 − 0.994i)14-s + (−0.683 + 0.729i)15-s + (0.996 − 0.0859i)16-s + (0.0215 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.579 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.579 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(293\)
Sign: $-0.579 + 0.815i$
Analytic conductor: \(1.36068\)
Root analytic conductor: \(1.36068\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{293} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 293,\ (0:\ ),\ -0.579 + 0.815i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1628870258 + 0.3156163390i\)
\(L(\frac12)\) \(\approx\) \(0.1628870258 + 0.3156163390i\)
\(L(1)\) \(\approx\) \(0.5786886000 + 0.1290216756i\)
\(L(1)\) \(\approx\) \(0.5786886000 + 0.1290216756i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad293 \( 1 \)
good2 \( 1 + (0.0215 + 0.999i)T \)
3 \( 1 + (-0.234 - 0.972i)T \)
5 \( 1 + (-0.548 - 0.835i)T \)
7 \( 1 + (-0.991 - 0.128i)T \)
11 \( 1 + (-0.618 + 0.785i)T \)
13 \( 1 + (0.869 - 0.493i)T \)
17 \( 1 + (0.0215 + 0.999i)T \)
19 \( 1 + (-0.683 + 0.729i)T \)
23 \( 1 + (0.966 + 0.255i)T \)
29 \( 1 + (0.357 - 0.933i)T \)
31 \( 1 + (-0.548 + 0.835i)T \)
37 \( 1 + (0.985 - 0.171i)T \)
41 \( 1 + (-0.925 + 0.377i)T \)
43 \( 1 + (0.192 + 0.981i)T \)
47 \( 1 + (-0.798 + 0.601i)T \)
53 \( 1 + (-0.618 + 0.785i)T \)
59 \( 1 + (-0.847 - 0.530i)T \)
61 \( 1 + (-0.954 - 0.296i)T \)
67 \( 1 + (0.436 - 0.899i)T \)
71 \( 1 + (0.908 - 0.417i)T \)
73 \( 1 + (-0.999 - 0.0430i)T \)
79 \( 1 + (-0.954 - 0.296i)T \)
83 \( 1 + (0.651 + 0.758i)T \)
89 \( 1 + (-0.954 + 0.296i)T \)
97 \( 1 + (-0.976 - 0.213i)T \)
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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

−25.72485754909553786184415540337, −23.609898662109048748464872160368, −23.12284698392443921391036749671, −22.20865425136495844224437031135, −21.66756289633014123802576812366, −20.69962893208956716007604870368, −19.78058316226070746575857066724, −18.82658393807010103471965347305, −18.278506604900300330033652097977, −16.809360716444496089825970649478, −15.95084397241605666544410509229, −15.0578954902106971643045652086, −13.94398892745965958268563434035, −13.02060420831094394325235938399, −11.67697138315144318683972850884, −11.043809096393636326687507863858, −10.35812730025772879085453442734, −9.28369030396534354692861754452, −8.500636853649427104242823487001, −6.76701371661692964092378490087, −5.56802350763477386546081150778, −4.32087678480234011747597492616, −3.30020918559277261907384913282, −2.75476188698134342189756385274, −0.26489464052788452902153473158, 1.288528433076681182723649411186, 3.33132311764424981605976818889, 4.61499167763418555367189742585, 5.808509277031809180899882713184, 6.5539190200221119195198080513, 7.74050203563739375494965643137, 8.28584536100095172554872273638, 9.43085817972386562135781836392, 10.759956759481952196128667677881, 12.449721614043776131213308635477, 12.81424246328338863137705356770, 13.49434261083027203380904298931, 14.92946901408442852423539997444, 15.786414088466854117901765039260, 16.68645619630046988645185548463, 17.36713483656994155015000995747, 18.42799793487403368065209767196, 19.22828341370804312088291660074, 20.063929393955652946562552708815, 21.38400711479157758293889999030, 22.9776663098567037772657190006, 23.1242652241814225117354421300, 23.83408255354423178376096851532, 25.06052826745906607381689435653, 25.38231410586294314372951579708

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