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Properties

Label 2-2793-2793.2180-c0-0-0
Degree $2$
Conductor $2793$
Sign $0.922 + 0.385i$
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  + (−0.980 + 0.198i)3-s + (−0.270 + 0.962i)4-s + (0.542 − 0.840i)7-s + (0.921 − 0.388i)9-s + (0.0747 − 0.997i)12-s + (0.636 + 0.0317i)13-s + (−0.853 − 0.521i)16-s + (0.222 − 0.974i)19-s + (−0.365 + 0.930i)21-s + (0.124 − 0.992i)25-s + (−0.826 + 0.563i)27-s + (0.661 + 0.749i)28-s − 1.93·31-s + (0.124 + 0.992i)36-s + (1.08 − 1.59i)37-s + ⋯
L(s)  = 1  + (−0.980 + 0.198i)3-s + (−0.270 + 0.962i)4-s + (0.542 − 0.840i)7-s + (0.921 − 0.388i)9-s + (0.0747 − 0.997i)12-s + (0.636 + 0.0317i)13-s + (−0.853 − 0.521i)16-s + (0.222 − 0.974i)19-s + (−0.365 + 0.930i)21-s + (0.124 − 0.992i)25-s + (−0.826 + 0.563i)27-s + (0.661 + 0.749i)28-s − 1.93·31-s + (0.124 + 0.992i)36-s + (1.08 − 1.59i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2793\)    =    \(3 \cdot 7^{2} \cdot 19\)
Sign: $0.922 + 0.385i$
Analytic conductor: \(1.39388\)
Root analytic conductor: \(1.18063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2793} (2180, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2793,\ (\ :0),\ 0.922 + 0.385i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8339362291\)
\(L(\frac12)\) \(\approx\) \(0.8339362291\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.980 - 0.198i)T \)
7 \( 1 + (-0.542 + 0.840i)T \)
19 \( 1 + (-0.222 + 0.974i)T \)
good2 \( 1 + (0.270 - 0.962i)T^{2} \)
5 \( 1 + (-0.124 + 0.992i)T^{2} \)
11 \( 1 + (-0.826 + 0.563i)T^{2} \)
13 \( 1 + (-0.636 - 0.0317i)T + (0.995 + 0.0995i)T^{2} \)
17 \( 1 + (-0.853 + 0.521i)T^{2} \)
23 \( 1 + (0.0249 - 0.999i)T^{2} \)
29 \( 1 + (0.980 - 0.198i)T^{2} \)
31 \( 1 + 1.93T + T^{2} \)
37 \( 1 + (-1.08 + 1.59i)T + (-0.365 - 0.930i)T^{2} \)
41 \( 1 + (0.124 - 0.992i)T^{2} \)
43 \( 1 + (-0.0459 + 1.84i)T + (-0.998 - 0.0498i)T^{2} \)
47 \( 1 + (0.995 + 0.0995i)T^{2} \)
53 \( 1 + (-0.853 - 0.521i)T^{2} \)
59 \( 1 + (-0.542 + 0.840i)T^{2} \)
61 \( 1 + (-0.0491 - 0.491i)T + (-0.980 + 0.198i)T^{2} \)
67 \( 1 + (-1.21 - 1.45i)T + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (-0.661 + 0.749i)T^{2} \)
73 \( 1 + (-1.27 - 0.653i)T + (0.583 + 0.811i)T^{2} \)
79 \( 1 + (-0.623 - 1.71i)T + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.0747 - 0.997i)T^{2} \)
89 \( 1 + (-0.270 - 0.962i)T^{2} \)
97 \( 1 + (1.37 - 0.501i)T + (0.766 - 0.642i)T^{2} \)
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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

−8.920563611207270525620761141383, −8.142160870330001531121584681027, −7.18232206404451035524621949586, −6.97194927915596395605240174877, −5.75233789968669131798530676574, −4.99877620191415591400316710017, −4.08261515862209721595071824226, −3.73694432512212714613567355278, −2.24310572280930145318567243242, −0.70132182927891253542825197811, 1.24868188878037167382302090498, 1.95500006417666435964194268237, 3.54013988646514979016592668277, 4.72904419198952320915193149711, 5.20371070758442315430413488461, 6.01591521895240884979290455733, 6.34915479228192493032173901219, 7.51905691990990773666212728411, 8.235376078627687899570051002470, 9.259383982642252335594656117193

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