↑

Properties

Label 2-1386-77.10-c1-0-16
Degree $2$
Conductor $1386$
Sign $0.951 + 0.307i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.725 + 0.418i)5-s + (−2.44 + 1.02i)7-s − 0.999i·8-s + (−0.418 + 0.725i)10-s + (3.15 + 1.01i)11-s + 2.59·13-s + (−1.60 + 2.10i)14-s + (−0.5 − 0.866i)16-s + (2.98 − 5.17i)17-s + (1.55 + 2.69i)19-s + 0.837i·20-s + (3.24 − 0.697i)22-s + (1.43 + 2.48i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.324 + 0.187i)5-s + (−0.922 + 0.386i)7-s − 0.353i·8-s + (−0.132 + 0.229i)10-s + (0.951 + 0.306i)11-s + 0.719·13-s + (−0.428 + 0.562i)14-s + (−0.125 − 0.216i)16-s + (0.724 − 1.25i)17-s + (0.356 + 0.618i)19-s + 0.187i·20-s + (0.691 − 0.148i)22-s + (0.298 + 0.517i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.951 + 0.307i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.951 + 0.307i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.264088860\)
\(L(\frac12)\) \(\approx\) \(2.264088860\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 + (2.44 - 1.02i)T \)
11 \( 1 + (-3.15 - 1.01i)T \)
good5 \( 1 + (0.725 - 0.418i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 2.59T + 13T^{2} \)
17 \( 1 + (-2.98 + 5.17i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.55 - 2.69i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.43 - 2.48i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.38iT - 29T^{2} \)
31 \( 1 + (0.913 + 0.527i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.49 - 9.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 + 1.27iT - 43T^{2} \)
47 \( 1 + (-10.6 + 6.12i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.58 + 4.48i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.38 - 4.84i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.03 + 3.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.51 - 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.0T + 71T^{2} \)
73 \( 1 + (-4.95 + 8.58i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-11.7 + 6.75i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.99T + 83T^{2} \)
89 \( 1 + (7.28 - 4.20i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.786iT - 97T^{2} \)
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

−9.580664638115628280675883636919, −9.022713674537053170648850108718, −7.73308030970341141800492421652, −6.99718743992697050580771322408, −6.08395969661804427912906860476, −5.46444562568044923181965449447, −4.16828699754306304637007630255, −3.49924285833389262790383764022, −2.59412272883456153827207113097, −1.08405255546372499088698776851, 1.02004246442751159557941315277, 2.77462269727824796610702292569, 3.85436117872530023324827130727, 4.20418730316526172026451023119, 5.70858297902353333692545451503, 6.19625218424456385067071496776, 7.05976714924081088063994666601, 7.85930712305634160790516705918, 8.827841423517200058653013324420, 9.415266385833649176832151967737

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