↑

Properties

Label 2-1386-33.29-c1-0-13
Degree $2$
Conductor $1386$
Sign $0.940 + 0.339i$
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  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.741 + 1.02i)5-s + (−0.951 + 0.309i)7-s + (0.309 − 0.951i)8-s − 1.26i·10-s + (3.31 − 0.0815i)11-s + (1.32 − 1.82i)13-s + (0.951 + 0.309i)14-s + (−0.809 + 0.587i)16-s + (3.80 − 2.76i)17-s + (−5.66 − 1.83i)19-s + (−0.741 + 1.02i)20-s + (−2.73 − 1.88i)22-s + 4.32i·23-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.154 + 0.475i)4-s + (0.331 + 0.456i)5-s + (−0.359 + 0.116i)7-s + (0.109 − 0.336i)8-s − 0.399i·10-s + (0.999 − 0.0245i)11-s + (0.368 − 0.507i)13-s + (0.254 + 0.0825i)14-s + (−0.202 + 0.146i)16-s + (0.922 − 0.670i)17-s + (−1.29 − 0.421i)19-s + (−0.165 + 0.228i)20-s + (−0.582 − 0.401i)22-s + 0.902i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.940 + 0.339i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (953, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.940 + 0.339i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.372183483\)
\(L(\frac12)\) \(\approx\) \(1.372183483\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 \)
7 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (-3.31 + 0.0815i)T \)
good5 \( 1 + (-0.741 - 1.02i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (-1.32 + 1.82i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.80 + 2.76i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (5.66 + 1.83i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 4.32iT - 23T^{2} \)
29 \( 1 + (-1.09 - 3.37i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.65 - 1.20i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.883 + 2.71i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.43 + 7.48i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 5.70iT - 43T^{2} \)
47 \( 1 + (-12.2 - 3.96i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.30 - 1.80i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-8.76 + 2.84i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.606 + 0.834i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 1.83T + 67T^{2} \)
71 \( 1 + (-6.53 - 8.99i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.28 + 0.418i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-8.05 + 11.0i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.78 - 2.02i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 6.74iT - 89T^{2} \)
97 \( 1 + (-2.44 - 1.77i)T + (29.9 + 92.2i)T^{2} \)
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

−9.456251236193367004745927058532, −8.961549168565997827931008680703, −8.035451345484183888718029550165, −7.08441559590451729923932552731, −6.40941861320700014044117704954, −5.50615447833874636597862687492, −4.16002605763207213277311671462, −3.24618413649425849009665282894, −2.29652296343753690394385663416, −0.927779765818445090114628912710, 0.998056897261072974429712199804, 2.11377321817820290226122321945, 3.68918690678258241706576197052, 4.53772061103366047285504506113, 5.79648376058524082168861115906, 6.34127590643400471120834513620, 7.11501761852450430010878114251, 8.252133901963303809463830276592, 8.738721297849844490763427140132, 9.532176756656564474526216315495

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