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Properties

Label 1-605-605.32-r0-0-0
Degree $1$
Conductor $605$
Sign $0.347 + 0.937i$
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  + (−0.281 + 0.959i)2-s − i·3-s + (−0.841 − 0.540i)4-s + (0.959 + 0.281i)6-s + (−0.909 − 0.415i)7-s + (0.755 − 0.654i)8-s − 9-s + (−0.540 + 0.841i)12-s + (−0.540 + 0.841i)13-s + (0.654 − 0.755i)14-s + (0.415 + 0.909i)16-s + (−0.989 + 0.142i)17-s + (0.281 − 0.959i)18-s + (−0.142 + 0.989i)19-s + (−0.415 + 0.909i)21-s + ⋯
L(s)  = 1  + (−0.281 + 0.959i)2-s − i·3-s + (−0.841 − 0.540i)4-s + (0.959 + 0.281i)6-s + (−0.909 − 0.415i)7-s + (0.755 − 0.654i)8-s − 9-s + (−0.540 + 0.841i)12-s + (−0.540 + 0.841i)13-s + (0.654 − 0.755i)14-s + (0.415 + 0.909i)16-s + (−0.989 + 0.142i)17-s + (0.281 − 0.959i)18-s + (−0.142 + 0.989i)19-s + (−0.415 + 0.909i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.347 + 0.937i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ 0.347 + 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5531908684 + 0.3848454828i\)
\(L(\frac12)\) \(\approx\) \(0.5531908684 + 0.3848454828i\)
\(L(1)\) \(\approx\) \(0.6771423301 + 0.1331872604i\)
\(L(1)\) \(\approx\) \(0.6771423301 + 0.1331872604i\)

Euler product

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

−22.531683010548264006521487347552, −22.059668671584115883838524247521, −21.24963447031523273517578500127, −20.478180553132190513260714058955, −19.512693630047384395361749476238, −19.27873775570723357218646780253, −17.6903216977644322126918310740, −17.426960121601812987027850301784, −16.167422505941842153126010849232, −15.51963396855265759930374959000, −14.55591961181377738821721073505, −13.35330987339987454914396804642, −12.75050020489787689514421831155, −11.626132086616590392249191177719, −10.92118602236425629902833778364, −10.0530359753590199771768551366, −9.30726374237163433695288544352, −8.76562952217739330096236097383, −7.51459358368439193397455540167, −6.06073434918627490036719098799, −4.967518004016579471219295289002, −4.16403107658710618252198055085, −2.96510851916012533022667189314, −2.5412085586115087325738546413, −0.476831618486539139862741587344, 0.96601303931366632349774806717, 2.32346365027550042673339473619, 3.77391320195253563299716652807, 4.94293502010080836490009220577, 6.18438174997719670348907058020, 6.70328025966131499477366786036, 7.416411283787925630524481130018, 8.44193880068536104663551997836, 9.24349767044627907408504243331, 10.20271046219430740679450175143, 11.35709592101510554340821032530, 12.63343556434730763714050956595, 13.14694532750874451176540754461, 14.07107941811418007292106774266, 14.72868599185053349580208463281, 15.8780158137602574898010734619, 16.77617401503851576755095350902, 17.22079879201376620629019146168, 18.27516642050439528072022671875, 19.02551474273577214661737170780, 19.49518228784051248734679627188, 20.49498798414732849058928126018, 22.021680972305454362155802223585, 22.69358683759152857603060250086, 23.426798401527668814947533701480

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