↑

Properties

Label 2-1386-77.54-c1-0-4
Degree $2$
Conductor $1386$
Sign $-0.997 + 0.0654i$
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 + (−1.09 − 0.629i)5-s + (−2.11 + 1.59i)7-s + 0.999i·8-s + (−0.629 − 1.09i)10-s + (−1.85 − 2.74i)11-s + 4.08·13-s + (−2.62 + 0.319i)14-s + (−0.5 + 0.866i)16-s + (−1.60 − 2.77i)17-s + (−3.81 + 6.60i)19-s − 1.25i·20-s + (−0.235 − 3.30i)22-s + (−4.12 + 7.14i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.487 − 0.281i)5-s + (−0.799 + 0.600i)7-s + 0.353i·8-s + (−0.199 − 0.344i)10-s + (−0.560 − 0.828i)11-s + 1.13·13-s + (−0.701 + 0.0854i)14-s + (−0.125 + 0.216i)16-s + (−0.388 − 0.672i)17-s + (−0.875 + 1.51i)19-s − 0.281i·20-s + (−0.0502 − 0.705i)22-s + (−0.860 + 1.49i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.997 + 0.0654i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.997 + 0.0654i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5126999857\)
\(L(\frac12)\) \(\approx\) \(0.5126999857\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.11 - 1.59i)T \)
11 \( 1 + (1.85 + 2.74i)T \)
good5 \( 1 + (1.09 + 0.629i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 4.08T + 13T^{2} \)
17 \( 1 + (1.60 + 2.77i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.81 - 6.60i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.12 - 7.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.54iT - 29T^{2} \)
31 \( 1 + (7.95 - 4.59i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.154 + 0.266i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.05T + 41T^{2} \)
43 \( 1 + 7.57iT - 43T^{2} \)
47 \( 1 + (4.07 + 2.35i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.39 + 4.15i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.36 - 1.36i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.755 + 1.30i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.69 + 2.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.50T + 71T^{2} \)
73 \( 1 + (0.483 + 0.837i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-13.5 - 7.80i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.32T + 83T^{2} \)
89 \( 1 + (9.22 + 5.32i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.6iT - 97T^{2} \)
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

−10.00997394156806352182650982831, −8.880038010534827459015222048126, −8.399707248082369883151344032966, −7.56690611631729557355660910939, −6.49274119614690799242272188367, −5.84086989704900001514187006821, −5.16351157765298070637149685147, −3.69945707274208373875059038192, −3.46962238405558134978580648979, −1.94184803751910906385030706313, 0.15588282057122799969612667176, 1.98167128025323380711466403170, 3.07768507931835947600913854043, 4.05934011479140956457775497990, 4.56969663285282069503556189958, 5.97981692337561557551463061012, 6.57587391731685250304723762542, 7.37812926479007564945743625878, 8.324733103736896748970921912852, 9.310007582272676521167212015923

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