↑

Properties

Label 2-156-156.35-c1-0-8
Degree $2$
Conductor $156$
Sign $0.691 + 0.722i$
Analytic cond. $1.24566$
Root an. cond. $1.11609$
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  + (−1.40 − 0.187i)2-s + (−1.29 − 1.15i)3-s + (1.92 + 0.526i)4-s + 1.98i·5-s + (1.59 + 1.86i)6-s + (2.42 − 1.40i)7-s + (−2.60 − 1.10i)8-s + (0.329 + 2.98i)9-s + (0.372 − 2.77i)10-s + (2.01 − 3.49i)11-s + (−1.88 − 2.90i)12-s + (0.235 − 3.59i)13-s + (−3.66 + 1.50i)14-s + (2.28 − 2.55i)15-s + (3.44 + 2.03i)16-s + (6.77 − 3.91i)17-s + ⋯
L(s)  = 1  + (−0.991 − 0.132i)2-s + (−0.744 − 0.667i)3-s + (0.964 + 0.263i)4-s + 0.885i·5-s + (0.649 + 0.760i)6-s + (0.916 − 0.529i)7-s + (−0.921 − 0.389i)8-s + (0.109 + 0.993i)9-s + (0.117 − 0.878i)10-s + (0.607 − 1.05i)11-s + (−0.542 − 0.839i)12-s + (0.0652 − 0.997i)13-s + (−0.978 + 0.402i)14-s + (0.591 − 0.659i)15-s + (0.861 + 0.508i)16-s + (1.64 − 0.948i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.691 + 0.722i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ 0.691 + 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.614664 - 0.262394i\)
\(L(\frac12)\) \(\approx\) \(0.614664 - 0.262394i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.187i)T \)
3 \( 1 + (1.29 + 1.15i)T \)
13 \( 1 + (-0.235 + 3.59i)T \)
good5 \( 1 - 1.98iT - 5T^{2} \)
7 \( 1 + (-2.42 + 1.40i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.01 + 3.49i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-6.77 + 3.91i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.36 - 1.94i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.939 - 1.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.74 - 1.00i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.09iT - 31T^{2} \)
37 \( 1 + (2.61 - 4.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.61 + 2.08i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.06 + 1.19i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.72T + 47T^{2} \)
53 \( 1 - 1.24iT - 53T^{2} \)
59 \( 1 + (-0.420 - 0.729i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.28 + 2.22i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.50 + 3.75i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.46 - 12.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.63T + 73T^{2} \)
79 \( 1 + 7.13iT - 79T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 + (-2.80 - 1.61i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.30 + 10.9i)T + (-48.5 + 84.0i)T^{2} \)
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

−12.39005182127676291334248662687, −11.54768347784688412369671841474, −10.78840790724306139887342326860, −10.15859985424364174383684792450, −8.381661541295101750358229194910, −7.61663575914321461092518953431, −6.66616027369180457963378791770, −5.52304356656319373588829490955, −3.15134974718966551144316863380, −1.17905980971576744727839667182, 1.59344417439248496181881021374, 4.32727953872839185401943739432, 5.48305341922811790755764319138, 6.68414572208709594371303594067, 8.159313031457381208285546927035, 9.077085591396145909623992925496, 9.884411928666069946598428930332, 10.97397178942128888094336475460, 11.99086422696734169133673753194, 12.40685136626726975140460620946

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