Properties

Label 1-1305-1305.992-r0-0-0
Degree $1$
Conductor $1305$
Sign $0.162 - 0.986i$
Analytic cond. $6.06039$
Root an. cond. $6.06039$
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  + (−0.680 − 0.733i)2-s + (−0.0747 + 0.997i)4-s + (−0.997 + 0.0747i)7-s + (0.781 − 0.623i)8-s + (−0.988 − 0.149i)11-s + (0.930 − 0.365i)13-s + (0.733 + 0.680i)14-s + (−0.988 − 0.149i)16-s + i·17-s + (−0.900 + 0.433i)19-s + (0.563 + 0.826i)22-s + (0.680 − 0.733i)23-s + (−0.900 − 0.433i)26-s − i·28-s + (−0.955 − 0.294i)31-s + (0.563 + 0.826i)32-s + ⋯
L(s)  = 1  + (−0.680 − 0.733i)2-s + (−0.0747 + 0.997i)4-s + (−0.997 + 0.0747i)7-s + (0.781 − 0.623i)8-s + (−0.988 − 0.149i)11-s + (0.930 − 0.365i)13-s + (0.733 + 0.680i)14-s + (−0.988 − 0.149i)16-s + i·17-s + (−0.900 + 0.433i)19-s + (0.563 + 0.826i)22-s + (0.680 − 0.733i)23-s + (−0.900 − 0.433i)26-s − i·28-s + (−0.955 − 0.294i)31-s + (0.563 + 0.826i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $0.162 - 0.986i$
Analytic conductor: \(6.06039\)
Root analytic conductor: \(6.06039\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (992, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1305,\ (0:\ ),\ 0.162 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5161403911 - 0.4380500017i\)
\(L(\frac12)\) \(\approx\) \(0.5161403911 - 0.4380500017i\)
\(L(1)\) \(\approx\) \(0.6023857015 - 0.1950519409i\)
\(L(1)\) \(\approx\) \(0.6023857015 - 0.1950519409i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.680 - 0.733i)T \)
7 \( 1 + (-0.997 + 0.0747i)T \)
11 \( 1 + (-0.988 - 0.149i)T \)
13 \( 1 + (0.930 - 0.365i)T \)
17 \( 1 + iT \)
19 \( 1 + (-0.900 + 0.433i)T \)
23 \( 1 + (0.680 - 0.733i)T \)
31 \( 1 + (-0.955 - 0.294i)T \)
37 \( 1 + (0.781 - 0.623i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.294 + 0.955i)T \)
47 \( 1 + (0.149 - 0.988i)T \)
53 \( 1 + (0.974 - 0.222i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.0747 - 0.997i)T \)
67 \( 1 + (-0.149 - 0.988i)T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (-0.974 - 0.222i)T \)
79 \( 1 + (0.365 - 0.930i)T \)
83 \( 1 + (-0.563 + 0.826i)T \)
89 \( 1 + (0.222 + 0.974i)T \)
97 \( 1 + (0.563 - 0.826i)T \)
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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

−20.95651837204775338529699228432, −20.25903933001677362061447771340, −19.385637245133755170035025310785, −18.71441463712862332455169473956, −18.213204256945197998462197907469, −17.245149430124786069717897427448, −16.52683699184429210523632465159, −15.75891117017752121829107582887, −15.44698001016430938485931625350, −14.33394551690024182865322109771, −13.40413391826290589054665352052, −13.011669141904139545404669537819, −11.656196175482825817506316939190, −10.75625533691883120509027008341, −10.157722945104663258807692963917, −9.15360671087180324062276276324, −8.766021108935522836848816734777, −7.54069988705631828250282690377, −7.01452922416651508680944125653, −6.11787350827992066545659258293, −5.37599451303589454806841409449, −4.3764227771722425949390280603, −3.15484699814809196002002207897, −2.11323444522424176781555432148, −0.781412598443378093888542958615, 0.47459339745662865288856334752, 1.78138885784958018389300963933, 2.76961364867919370290136424620, 3.510086824123764913404168326426, 4.365245220370549309697989677248, 5.75269847953956681920604126939, 6.51570302010511728902478203159, 7.590228726143971805667272551215, 8.40554598297516767885364720454, 9.00276199244727426165495279226, 10.06138065406419963067970510156, 10.582191139947321453714251223567, 11.223913920179112942792348183930, 12.444103110875979750146459923218, 12.9773420938071040180448095225, 13.36914755973374118216270955025, 14.79716403779059554510230304282, 15.63739711945729538025240395917, 16.4960309324902044889046093864, 16.88816590437642532643129142982, 18.14009426538292820857718202042, 18.47951282388698932737719303132, 19.29866461889552255442051242671, 19.92094575549467094023928948297, 20.78190331283018975798189494419

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