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Properties

Label 1-2793-2793.1871-r1-0-0
Degree $1$
Conductor $2793$
Sign $0.948 + 0.316i$
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  + (0.411 + 0.911i)2-s + (−0.661 + 0.749i)4-s + (0.998 − 0.0498i)5-s + (−0.955 − 0.294i)8-s + (0.456 + 0.889i)10-s + (−0.0747 − 0.997i)11-s + (0.270 − 0.962i)13-s + (−0.124 − 0.992i)16-s + (0.661 + 0.749i)17-s + (−0.623 + 0.781i)20-s + (0.878 − 0.478i)22-s + (0.318 + 0.947i)23-s + (0.995 − 0.0995i)25-s + (0.988 − 0.149i)26-s + (0.853 − 0.521i)29-s + ⋯
L(s)  = 1  + (0.411 + 0.911i)2-s + (−0.661 + 0.749i)4-s + (0.998 − 0.0498i)5-s + (−0.955 − 0.294i)8-s + (0.456 + 0.889i)10-s + (−0.0747 − 0.997i)11-s + (0.270 − 0.962i)13-s + (−0.124 − 0.992i)16-s + (0.661 + 0.749i)17-s + (−0.623 + 0.781i)20-s + (0.878 − 0.478i)22-s + (0.318 + 0.947i)23-s + (0.995 − 0.0995i)25-s + (0.988 − 0.149i)26-s + (0.853 − 0.521i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2793\)    =    \(3 \cdot 7^{2} \cdot 19\)
Sign: $0.948 + 0.316i$
Analytic conductor: \(300.149\)
Root analytic conductor: \(300.149\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2793} (1871, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2793,\ (1:\ ),\ 0.948 + 0.316i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.311793504 + 0.5380728097i\)
\(L(\frac12)\) \(\approx\) \(3.311793504 + 0.5380728097i\)
\(L(1)\) \(\approx\) \(1.418573130 + 0.5431431342i\)
\(L(1)\) \(\approx\) \(1.418573130 + 0.5431431342i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.411 + 0.911i)T \)
5 \( 1 + (0.998 - 0.0498i)T \)
11 \( 1 + (-0.0747 - 0.997i)T \)
13 \( 1 + (0.270 - 0.962i)T \)
17 \( 1 + (0.661 + 0.749i)T \)
23 \( 1 + (0.318 + 0.947i)T \)
29 \( 1 + (0.853 - 0.521i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.988 - 0.149i)T \)
41 \( 1 + (0.998 - 0.0498i)T \)
43 \( 1 + (-0.797 + 0.603i)T \)
47 \( 1 + (-0.270 + 0.962i)T \)
53 \( 1 + (0.661 - 0.749i)T \)
59 \( 1 + (-0.921 - 0.388i)T \)
61 \( 1 + (-0.853 + 0.521i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.0249 - 0.999i)T \)
73 \( 1 + (-0.969 - 0.246i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (-0.826 - 0.563i)T \)
89 \( 1 + (0.411 - 0.911i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−18.83207603879305656981965619392, −18.54112102045395212504896361118, −17.73689987443436520566094063363, −17.11197711347534553244762357605, −16.22217404996306380341414052030, −15.252201126419873954293509266222, −14.45425959127976979257583012786, −13.96109517647202323273863620694, −13.39475677100926585488432302229, −12.44674614884793351752319103444, −12.077833057092013788820209108203, −11.1503175233170804926283565667, −10.24820355240728339899251997987, −9.942275590390756558465542084198, −9.09744526169621211817062941519, −8.512519491447206296390868444015, −7.08702534573933742306048995806, −6.51324809991859839471494274257, −5.57279451803209872416763043745, −4.8421036525236903701203243721, −4.266273588384164305251338495217, −3.08258575723678992834273182095, −2.43002796628505523588601218130, −1.64684808685285773711582341463, −0.88392915645759436412436188733, 0.52991904228297840670164164513, 1.486759454636116269993019372040, 2.925859499826986324545367124308, 3.29830848055294997818658273177, 4.47669841255995585961386493731, 5.330506084238398842810968151966, 5.94251860145764805147100576988, 6.331420232053596785745253844935, 7.43194275598035907952327321457, 8.21287672332516815587582501991, 8.72718732408769869634130079500, 9.6745653732693572427635396267, 10.311535204860317483700761207365, 11.23260521074238909675725135698, 12.263046880749544691333272189805, 12.92782427859681822825990125500, 13.59317607789817242102055457349, 14.024765212345246369735648751200, 14.836298759448883048863447084, 15.578402150643323359789168704243, 16.21477996320251364919859589021, 17.04309893062589020583547263980, 17.48574494163093723002454133236, 18.101864711364614307049291486111, 18.92494792366274226765302543536

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