Properties

Label 2-1386-77.76-c1-0-9
Degree $2$
Conductor $1386$
Sign $-0.0230 - 0.999i$
Analytic cond. $11.0672$
Root an. cond. $3.32675$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 4-s + 0.266i·5-s + (−1.34 − 2.27i)7-s i·8-s − 0.266·10-s + (−1.62 + 2.89i)11-s + 2.55·13-s + (2.27 − 1.34i)14-s + 16-s + 4.96·17-s − 8.21·19-s − 0.266i·20-s + (−2.89 − 1.62i)22-s + 5.23·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.118i·5-s + (−0.509 − 0.860i)7-s − 0.353i·8-s − 0.0841·10-s + (−0.489 + 0.871i)11-s + 0.707·13-s + (0.608 − 0.360i)14-s + 0.250·16-s + 1.20·17-s − 1.88·19-s − 0.0594i·20-s + (−0.616 − 0.346i)22-s + 1.09·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0230 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0230 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.0230 - 0.999i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.0230 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.347519432\)
\(L(\frac12)\) \(\approx\) \(1.347519432\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 + (1.34 + 2.27i)T \)
11 \( 1 + (1.62 - 2.89i)T \)
good5 \( 1 - 0.266iT - 5T^{2} \)
13 \( 1 - 2.55T + 13T^{2} \)
17 \( 1 - 4.96T + 17T^{2} \)
19 \( 1 + 8.21T + 19T^{2} \)
23 \( 1 - 5.23T + 23T^{2} \)
29 \( 1 - 5.25iT - 29T^{2} \)
31 \( 1 - 4.28iT - 31T^{2} \)
37 \( 1 - 7.39T + 37T^{2} \)
41 \( 1 + 6.21T + 41T^{2} \)
43 \( 1 + 1.46iT - 43T^{2} \)
47 \( 1 - 10.0iT - 47T^{2} \)
53 \( 1 - 13.1T + 53T^{2} \)
59 \( 1 - 9.10iT - 59T^{2} \)
61 \( 1 - 5.98T + 61T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 + 4.93T + 71T^{2} \)
73 \( 1 - 2.96T + 73T^{2} \)
79 \( 1 - 6.16iT - 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 - 1.78iT - 89T^{2} \)
97 \( 1 + 0.146iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.786998060405489556092680289908, −8.814202516952155555188077946183, −8.134845304295891980507560912786, −7.07749648318206365182296300300, −6.79835164178119195644142286985, −5.72307694664673083719129475598, −4.75645492548475997268043059182, −3.92583394437717635241497649678, −2.86059656683683872351476360127, −1.12845930899476964938928448555, 0.66236665493023983087966835817, 2.23411306119386551427502474238, 3.10241737984337552151508876162, 4.00958612522419509390573386076, 5.22963131284920749960299203636, 5.91055757170386449439546755215, 6.77525345473457783706740015889, 8.281052958505975349670542640627, 8.477973024907906035802468858542, 9.427453471394159860192040845293

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