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Properties

Label 2-2793-2793.1346-c0-0-0
Degree $2$
Conductor $2793$
Sign $0.457 + 0.889i$
Analytic cond. $1.39388$
Root an. cond. $1.18063$
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  + (−0.797 + 0.603i)3-s + (−0.0249 − 0.999i)4-s + (0.698 − 0.715i)7-s + (0.270 − 0.962i)9-s + (0.623 + 0.781i)12-s + (0.907 − 0.0908i)13-s + (−0.998 + 0.0498i)16-s + (0.0747 + 0.997i)19-s + (−0.124 + 0.992i)21-s + (0.270 − 0.962i)25-s + (0.365 + 0.930i)27-s + (−0.733 − 0.680i)28-s + (0.661 − 1.14i)31-s + (−0.969 − 0.246i)36-s + (0.629 − 0.0949i)37-s + ⋯
L(s)  = 1  + (−0.797 + 0.603i)3-s + (−0.0249 − 0.999i)4-s + (0.698 − 0.715i)7-s + (0.270 − 0.962i)9-s + (0.623 + 0.781i)12-s + (0.907 − 0.0908i)13-s + (−0.998 + 0.0498i)16-s + (0.0747 + 0.997i)19-s + (−0.124 + 0.992i)21-s + (0.270 − 0.962i)25-s + (0.365 + 0.930i)27-s + (−0.733 − 0.680i)28-s + (0.661 − 1.14i)31-s + (−0.969 − 0.246i)36-s + (0.629 − 0.0949i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2793\)    =    \(3 \cdot 7^{2} \cdot 19\)
Sign: $0.457 + 0.889i$
Analytic conductor: \(1.39388\)
Root analytic conductor: \(1.18063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2793} (1346, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2793,\ (\ :0),\ 0.457 + 0.889i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9997387459\)
\(L(\frac12)\) \(\approx\) \(0.9997387459\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.797 - 0.603i)T \)
7 \( 1 + (-0.698 + 0.715i)T \)
19 \( 1 + (-0.0747 - 0.997i)T \)
good2 \( 1 + (0.0249 + 0.999i)T^{2} \)
5 \( 1 + (-0.270 + 0.962i)T^{2} \)
11 \( 1 + (-0.623 + 0.781i)T^{2} \)
13 \( 1 + (-0.907 + 0.0908i)T + (0.980 - 0.198i)T^{2} \)
17 \( 1 + (-0.542 + 0.840i)T^{2} \)
23 \( 1 + (-0.456 + 0.889i)T^{2} \)
29 \( 1 + (-0.921 - 0.388i)T^{2} \)
31 \( 1 + (-0.661 + 1.14i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.629 + 0.0949i)T + (0.955 - 0.294i)T^{2} \)
41 \( 1 + (0.969 - 0.246i)T^{2} \)
43 \( 1 + (1.98 - 0.0991i)T + (0.995 - 0.0995i)T^{2} \)
47 \( 1 + (0.318 - 0.947i)T^{2} \)
53 \( 1 + (0.998 - 0.0498i)T^{2} \)
59 \( 1 + (-0.995 + 0.0995i)T^{2} \)
61 \( 1 + (-0.421 - 1.25i)T + (-0.797 + 0.603i)T^{2} \)
67 \( 1 + (0.857 + 0.312i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (0.124 - 0.992i)T^{2} \)
73 \( 1 + (0.806 + 1.78i)T + (-0.661 + 0.749i)T^{2} \)
79 \( 1 + (-0.305 + 1.72i)T + (-0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.988 - 0.149i)T^{2} \)
89 \( 1 + (-0.878 + 0.478i)T^{2} \)
97 \( 1 + (-1.26 - 1.06i)T + (0.173 + 0.984i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.954208257229999548377751228109, −8.182363609322613870370279902890, −7.20872902241092365603312966982, −6.17253316403591295378735098556, −5.96578199609357901340766801327, −4.84273148212416096469176815187, −4.41162624280751203948951118893, −3.45524475264355129522969377385, −1.80043595287506752746670873125, −0.798792268970554864845252564369, 1.37743466715932231421529577595, 2.46188600174208019343550371977, 3.43582733237908922080763361917, 4.64863265372302125525436254570, 5.16157476158867050173275997025, 6.17810559761934592645806407984, 6.88747010663576540343950947427, 7.54592098685592540581246966490, 8.489299611652349598082946508204, 8.671995038444861525846502440164

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