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Properties

Label 2-2793-2793.1871-c0-0-0
Degree $2$
Conductor $2793$
Sign $0.151 - 0.988i$
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  + (−0.853 + 0.521i)3-s + (−0.411 − 0.911i)4-s + (0.921 + 0.388i)7-s + (0.456 − 0.889i)9-s + (0.826 + 0.563i)12-s + (−1.40 + 1.06i)13-s + (−0.661 + 0.749i)16-s + (−0.222 + 0.974i)19-s + (−0.988 + 0.149i)21-s + (−0.998 + 0.0498i)25-s + (0.0747 + 0.997i)27-s + (−0.0249 − 0.999i)28-s + 1.99·31-s + (−0.998 − 0.0498i)36-s + (−0.0614 + 0.820i)37-s + ⋯
L(s)  = 1  + (−0.853 + 0.521i)3-s + (−0.411 − 0.911i)4-s + (0.921 + 0.388i)7-s + (0.456 − 0.889i)9-s + (0.826 + 0.563i)12-s + (−1.40 + 1.06i)13-s + (−0.661 + 0.749i)16-s + (−0.222 + 0.974i)19-s + (−0.988 + 0.149i)21-s + (−0.998 + 0.0498i)25-s + (0.0747 + 0.997i)27-s + (−0.0249 − 0.999i)28-s + 1.99·31-s + (−0.998 − 0.0498i)36-s + (−0.0614 + 0.820i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2793\)    =    \(3 \cdot 7^{2} \cdot 19\)
Sign: $0.151 - 0.988i$
Analytic conductor: \(1.39388\)
Root analytic conductor: \(1.18063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2793} (1871, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2793,\ (\ :0),\ 0.151 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6637333396\)
\(L(\frac12)\) \(\approx\) \(0.6637333396\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.853 - 0.521i)T \)
7 \( 1 + (-0.921 - 0.388i)T \)
19 \( 1 + (0.222 - 0.974i)T \)
good2 \( 1 + (0.411 + 0.911i)T^{2} \)
5 \( 1 + (0.998 - 0.0498i)T^{2} \)
11 \( 1 + (-0.0747 - 0.997i)T^{2} \)
13 \( 1 + (1.40 - 1.06i)T + (0.270 - 0.962i)T^{2} \)
17 \( 1 + (0.661 + 0.749i)T^{2} \)
23 \( 1 + (0.318 + 0.947i)T^{2} \)
29 \( 1 + (0.853 - 0.521i)T^{2} \)
31 \( 1 - 1.99T + T^{2} \)
37 \( 1 + (0.0614 - 0.820i)T + (-0.988 - 0.149i)T^{2} \)
41 \( 1 + (0.998 - 0.0498i)T^{2} \)
43 \( 1 + (0.290 + 0.864i)T + (-0.797 + 0.603i)T^{2} \)
47 \( 1 + (-0.270 + 0.962i)T^{2} \)
53 \( 1 + (0.661 - 0.749i)T^{2} \)
59 \( 1 + (-0.921 - 0.388i)T^{2} \)
61 \( 1 + (-0.538 - 1.91i)T + (-0.853 + 0.521i)T^{2} \)
67 \( 1 + (1.65 - 0.600i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (0.0249 - 0.999i)T^{2} \)
73 \( 1 + (-0.145 + 1.15i)T + (-0.969 - 0.246i)T^{2} \)
79 \( 1 + (-0.242 - 1.37i)T + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.826 - 0.563i)T^{2} \)
89 \( 1 + (0.411 - 0.911i)T^{2} \)
97 \( 1 + (-0.331 - 1.88i)T + (-0.939 + 0.342i)T^{2} \)
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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

−9.301894574867595520952729758009, −8.595767737185467204881296926874, −7.61569375761722839196269698668, −6.63630715170682483813468291878, −5.96040066318992438039005604538, −5.19763152052003472455538525086, −4.62561772435136773494467045156, −4.03217696881711834384166891841, −2.33086654544292113190713202791, −1.31447572587706667970429273326, 0.50788688657082750301010326352, 2.09262655952713429780981462467, 3.03341470744638785089300268938, 4.46755945838327533529780667199, 4.74360400980910268846322355753, 5.62982701549187073091657915331, 6.70468997813687231081478024540, 7.48043094337997840280337811723, 7.86270164913274988029060994616, 8.520869004771327215033798920723

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