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Properties

Label 2-2793-2793.584-c0-0-0
Degree $2$
Conductor $2793$
Sign $0.252 + 0.967i$
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 + (−0.878 − 0.478i)4-s + (0.995 − 0.0995i)7-s + (0.270 − 0.962i)9-s + (−0.988 + 0.149i)12-s + (0.145 + 0.201i)13-s + (0.542 + 0.840i)16-s + (0.900 − 0.433i)19-s + (0.733 − 0.680i)21-s + (−0.698 − 0.715i)25-s + (−0.365 − 0.930i)27-s + (−0.921 − 0.388i)28-s − 0.0498·31-s + (−0.698 + 0.715i)36-s + (0.890 + 0.349i)37-s + ⋯
L(s)  = 1  + (0.797 − 0.603i)3-s + (−0.878 − 0.478i)4-s + (0.995 − 0.0995i)7-s + (0.270 − 0.962i)9-s + (−0.988 + 0.149i)12-s + (0.145 + 0.201i)13-s + (0.542 + 0.840i)16-s + (0.900 − 0.433i)19-s + (0.733 − 0.680i)21-s + (−0.698 − 0.715i)25-s + (−0.365 − 0.930i)27-s + (−0.921 − 0.388i)28-s − 0.0498·31-s + (−0.698 + 0.715i)36-s + (0.890 + 0.349i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2793\)    =    \(3 \cdot 7^{2} \cdot 19\)
Sign: $0.252 + 0.967i$
Analytic conductor: \(1.39388\)
Root analytic conductor: \(1.18063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2793} (584, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2793,\ (\ :0),\ 0.252 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.496817628\)
\(L(\frac12)\) \(\approx\) \(1.496817628\)
\(L(1)\) 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 \( 1 + (-0.995 + 0.0995i)T \)
19 \( 1 + (-0.900 + 0.433i)T \)
good2 \( 1 + (0.878 + 0.478i)T^{2} \)
5 \( 1 + (0.698 + 0.715i)T^{2} \)
11 \( 1 + (-0.365 - 0.930i)T^{2} \)
13 \( 1 + (-0.145 - 0.201i)T + (-0.318 + 0.947i)T^{2} \)
17 \( 1 + (0.542 - 0.840i)T^{2} \)
23 \( 1 + (-0.456 + 0.889i)T^{2} \)
29 \( 1 + (-0.797 + 0.603i)T^{2} \)
31 \( 1 + 0.0498T + T^{2} \)
37 \( 1 + (-0.890 - 0.349i)T + (0.733 + 0.680i)T^{2} \)
41 \( 1 + (-0.698 - 0.715i)T^{2} \)
43 \( 1 + (0.247 - 0.482i)T + (-0.583 - 0.811i)T^{2} \)
47 \( 1 + (-0.318 + 0.947i)T^{2} \)
53 \( 1 + (0.542 + 0.840i)T^{2} \)
59 \( 1 + (-0.995 + 0.0995i)T^{2} \)
61 \( 1 + (1.89 - 0.636i)T + (0.797 - 0.603i)T^{2} \)
67 \( 1 + (1.27 + 1.52i)T + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.921 - 0.388i)T^{2} \)
73 \( 1 + (-0.950 + 0.428i)T + (0.661 - 0.749i)T^{2} \)
79 \( 1 + (-0.135 - 0.372i)T + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (-0.988 + 0.149i)T^{2} \)
89 \( 1 + (-0.878 + 0.478i)T^{2} \)
97 \( 1 + (-0.140 + 0.0511i)T + (0.766 - 0.642i)T^{2} \)
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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

−8.866782828014789649460834369672, −7.979789484851636752599735739440, −7.68084672051546848370696664527, −6.55656191886376751339482362450, −5.77042532267771542208078961443, −4.78598847315277647855069613603, −4.16865741175530909167248554579, −3.12692273044954609262036400063, −1.94014339088109198051324721988, −1.02569329733898939482475152234, 1.53026188493785237850534419077, 2.78154784859434806254227044657, 3.65309276393886873978996082463, 4.33695780715441838503783617676, 5.09902578678749943418193988980, 5.75182960840384803649559402097, 7.35742436580405980369038368383, 7.79845754254051663150662272460, 8.411024771230598672487775277122, 9.113865886549118118564407796875

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