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Properties

Label 1-605-605.593-r0-0-0
Degree $1$
Conductor $605$
Sign $-0.714 + 0.699i$
Analytic cond. $2.80960$
Root an. cond. $2.80960$
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  + (0.909 + 0.415i)2-s + i·3-s + (0.654 + 0.755i)4-s + (−0.415 + 0.909i)6-s + (0.989 − 0.142i)7-s + (0.281 + 0.959i)8-s − 9-s + (−0.755 + 0.654i)12-s + (−0.755 + 0.654i)13-s + (0.959 + 0.281i)14-s + (−0.142 + 0.989i)16-s + (−0.540 + 0.841i)17-s + (−0.909 − 0.415i)18-s + (0.841 − 0.540i)19-s + (0.142 + 0.989i)21-s + ⋯
L(s)  = 1  + (0.909 + 0.415i)2-s + i·3-s + (0.654 + 0.755i)4-s + (−0.415 + 0.909i)6-s + (0.989 − 0.142i)7-s + (0.281 + 0.959i)8-s − 9-s + (−0.755 + 0.654i)12-s + (−0.755 + 0.654i)13-s + (0.959 + 0.281i)14-s + (−0.142 + 0.989i)16-s + (−0.540 + 0.841i)17-s + (−0.909 − 0.415i)18-s + (0.841 − 0.540i)19-s + (0.142 + 0.989i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.714 + 0.699i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ -0.714 + 0.699i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9241090582 + 2.266602595i\)
\(L(\frac12)\) \(\approx\) \(0.9241090582 + 2.266602595i\)
\(L(1)\) \(\approx\) \(1.367892006 + 1.187878700i\)
\(L(1)\) \(\approx\) \(1.367892006 + 1.187878700i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.909 + 0.415i)T \)
3 \( 1 + iT \)
7 \( 1 + (0.989 - 0.142i)T \)
13 \( 1 + (-0.755 + 0.654i)T \)
17 \( 1 + (-0.540 + 0.841i)T \)
19 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (-0.989 - 0.142i)T \)
29 \( 1 + (0.841 - 0.540i)T \)
31 \( 1 + (-0.654 + 0.755i)T \)
37 \( 1 + (0.755 + 0.654i)T \)
41 \( 1 + (-0.415 + 0.909i)T \)
43 \( 1 + (0.281 + 0.959i)T \)
47 \( 1 + (0.909 - 0.415i)T \)
53 \( 1 + (0.989 - 0.142i)T \)
59 \( 1 + (-0.415 - 0.909i)T \)
61 \( 1 + (-0.415 - 0.909i)T \)
67 \( 1 + (-0.909 - 0.415i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (0.989 + 0.142i)T \)
79 \( 1 + (-0.959 - 0.281i)T \)
83 \( 1 + (-0.989 + 0.142i)T \)
89 \( 1 + (-0.841 - 0.540i)T \)
97 \( 1 + (0.281 + 0.959i)T \)
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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

−22.64918384069022614656946406796, −22.183276635632774721186025394912, −21.07263337078466125444886534837, −20.13171974625442420142675833130, −19.85077854624522797548312138303, −18.51557517538221192631032842704, −18.098412750380223286280975346315, −17.045830592755463634037699047, −15.86000567914297465788448762397, −14.883457145827347952928747577304, −14.10948961120589187186357977001, −13.59934428192106812980736612380, −12.44302730087862311481872300069, −11.96752615573893626472922908165, −11.18813141462233634071928016469, −10.20179149595071381439540042660, −8.92621881384965643549385844248, −7.65470345194348848603539396150, −7.17941997469742046959418712215, −5.80889788481054180549888591497, −5.29573717889442741540936836126, −4.11822774192998854963648052319, −2.75835869876348480783381570172, −2.07120948092099509658820013861, −0.91424517729250194253351976334, 1.93989942497444471139544892794, 3.02999748241522845039972335489, 4.25063300000622210884456619971, 4.66876829124890031871279199711, 5.61124116210380872777281706965, 6.65539190590167844473948268957, 7.82666343463869674691224262632, 8.57011629016666822672841775894, 9.74578624661222781316275292194, 10.83806893811266995670545431152, 11.530812935269259062740201124575, 12.278976475926308842752984793005, 13.645393659703045789022050877603, 14.26578726405894130985568845992, 14.974324051915833889340616269161, 15.68111796068296451348015218515, 16.58833240734140682333388785246, 17.26033908433317172082479594006, 18.07102731879198997258778902291, 19.86432074483508749854661072581, 20.13622831443376527140672665391, 21.42848643510129564584441910653, 21.57406092507384256326983262119, 22.39786472642548386151355444437, 23.424985296498477810499397928097

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