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Properties

Label 1-2793-2793.269-r1-0-0
Degree $1$
Conductor $2793$
Sign $-0.609 + 0.792i$
Analytic cond. $300.149$
Root an. cond. $300.149$
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  + (0.698 + 0.715i)2-s + (−0.0249 + 0.999i)4-s + (−0.797 − 0.603i)5-s + (−0.733 + 0.680i)8-s + (−0.124 − 0.992i)10-s + (−0.826 + 0.563i)11-s + (−0.411 − 0.911i)13-s + (−0.998 − 0.0498i)16-s + (−0.0249 − 0.999i)17-s + (0.623 − 0.781i)20-s + (−0.980 − 0.198i)22-s + (−0.878 + 0.478i)23-s + (0.270 + 0.962i)25-s + (0.365 − 0.930i)26-s + (−0.661 − 0.749i)29-s + ⋯
L(s)  = 1  + (0.698 + 0.715i)2-s + (−0.0249 + 0.999i)4-s + (−0.797 − 0.603i)5-s + (−0.733 + 0.680i)8-s + (−0.124 − 0.992i)10-s + (−0.826 + 0.563i)11-s + (−0.411 − 0.911i)13-s + (−0.998 − 0.0498i)16-s + (−0.0249 − 0.999i)17-s + (0.623 − 0.781i)20-s + (−0.980 − 0.198i)22-s + (−0.878 + 0.478i)23-s + (0.270 + 0.962i)25-s + (0.365 − 0.930i)26-s + (−0.661 − 0.749i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2793\)    =    \(3 \cdot 7^{2} \cdot 19\)
Sign: $-0.609 + 0.792i$
Analytic conductor: \(300.149\)
Root analytic conductor: \(300.149\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2793} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2793,\ (1:\ ),\ -0.609 + 0.792i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3919027174 + 0.7962483159i\)
\(L(\frac12)\) \(\approx\) \(0.3919027174 + 0.7962483159i\)
\(L(1)\) \(\approx\) \(0.9576715986 + 0.3118074580i\)
\(L(1)\) \(\approx\) \(0.9576715986 + 0.3118074580i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.698 + 0.715i)T \)
5 \( 1 + (-0.797 - 0.603i)T \)
11 \( 1 + (-0.826 + 0.563i)T \)
13 \( 1 + (-0.411 - 0.911i)T \)
17 \( 1 + (-0.0249 - 0.999i)T \)
23 \( 1 + (-0.878 + 0.478i)T \)
29 \( 1 + (-0.661 - 0.749i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.365 - 0.930i)T \)
41 \( 1 + (0.797 + 0.603i)T \)
43 \( 1 + (0.542 - 0.840i)T \)
47 \( 1 + (-0.411 - 0.911i)T \)
53 \( 1 + (-0.0249 + 0.999i)T \)
59 \( 1 + (-0.456 - 0.889i)T \)
61 \( 1 + (0.661 + 0.749i)T \)
67 \( 1 + (0.939 - 0.342i)T \)
71 \( 1 + (-0.318 - 0.947i)T \)
73 \( 1 + (-0.995 + 0.0995i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (0.0747 - 0.997i)T \)
89 \( 1 + (-0.698 + 0.715i)T \)
97 \( 1 + (0.173 + 0.984i)T \)
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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

−19.04334825371103919327224726130, −18.421972441958739247062197679185, −17.57945725545567921955685334088, −16.40778088895435493251641516513, −15.8375089546724964441505955777, −15.13414594745393559568036865639, −14.38233749037953151286970939921, −13.97674064639882519262806574444, −12.93558205927328434583331888337, −12.4219136117648127539351889413, −11.57507959020928167946578238452, −11.08767262956295828189975652682, −10.38204772566571404855082041529, −9.74839632245282815366247599396, −8.63382884556948299148933394779, −7.95209470843418096186334784273, −6.92627213199225228599724701766, −6.28089039398505633929893536671, −5.44232888630684737970443758870, −4.43898913370884923520612525814, −3.95402248564641739972120771566, −3.02618986984849759116615168178, −2.40968794099538150119575595182, −1.38721756787496317837220556683, −0.17688313209830646630163667393, 0.56422197647986622662392205030, 2.17324331261570390266725947556, 2.98423282706431328115280001835, 3.88128186142572547187000898771, 4.623311244549181967645744626179, 5.25629775443684493469813940376, 5.89790226174787953360642680899, 7.09219222731913348706244536840, 7.67816902589039751436990238196, 8.05317887594199829599192692562, 9.032008167891345226614905420563, 9.83776479792218341140472253882, 10.86949971130740828400782282765, 11.83600891171992271420880058297, 12.21007838164787319043948264465, 13.023760564326874785147392882358, 13.51658720911325318104286651501, 14.4606019922036710017769611144, 15.24204792931347526000369492987, 15.74325282022839505086827983697, 16.15744724266755292766525347165, 17.12152990326139163778625638664, 17.687258866877863705193104133739, 18.40413226795666300020613031101, 19.396814171505561749214917940029

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