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Properties

Label 2-156-156.107-c1-0-11
Degree $2$
Conductor $156$
Sign $0.813 + 0.582i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.226 + 1.39i)2-s + (−1.71 − 0.207i)3-s + (−1.89 − 0.632i)4-s − 3.37i·5-s + (0.679 − 2.35i)6-s + (1.77 + 1.02i)7-s + (1.31 − 2.50i)8-s + (2.91 + 0.713i)9-s + (4.70 + 0.763i)10-s + (−2.01 − 3.48i)11-s + (3.13 + 1.48i)12-s + (−0.274 − 3.59i)13-s + (−1.83 + 2.24i)14-s + (−0.699 + 5.79i)15-s + (3.19 + 2.40i)16-s + (−0.0707 − 0.0408i)17-s + ⋯
L(s)  = 1  + (−0.160 + 0.987i)2-s + (−0.992 − 0.119i)3-s + (−0.948 − 0.316i)4-s − 1.50i·5-s + (0.277 − 0.960i)6-s + (0.670 + 0.387i)7-s + (0.464 − 0.885i)8-s + (0.971 + 0.237i)9-s + (1.48 + 0.241i)10-s + (−0.607 − 1.05i)11-s + (0.903 + 0.427i)12-s + (−0.0760 − 0.997i)13-s + (−0.489 + 0.599i)14-s + (−0.180 + 1.49i)15-s + (0.799 + 0.600i)16-s + (−0.0171 − 0.00991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.813 + 0.582i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ 0.813 + 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.632989 - 0.203207i\)
\(L(\frac12)\) \(\approx\) \(0.632989 - 0.203207i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.226 - 1.39i)T \)
3 \( 1 + (1.71 + 0.207i)T \)
13 \( 1 + (0.274 + 3.59i)T \)
good5 \( 1 + 3.37iT - 5T^{2} \)
7 \( 1 + (-1.77 - 1.02i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.01 + 3.48i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.0707 + 0.0408i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.648 + 0.374i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.176 + 0.306i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.448 + 0.258i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.36iT - 31T^{2} \)
37 \( 1 + (-5.25 - 9.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.18 - 2.99i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.33 - 2.50i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.77T + 47T^{2} \)
53 \( 1 - 7.94iT - 53T^{2} \)
59 \( 1 + (4.06 - 7.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.63 - 2.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.16 + 4.71i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.29 + 2.24i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 - 6.08iT - 79T^{2} \)
83 \( 1 - 1.15T + 83T^{2} \)
89 \( 1 + (-6.78 + 3.91i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.60 + 13.1i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.99928162414522407991128288890, −12.02005098614047916135682177250, −10.81857038655776146698239934922, −9.590881289251931756240286237387, −8.388766892673278768609811222483, −7.80938912099304674081059917596, −6.03574889193882953132002095789, −5.35530847172423594914577942833, −4.51009780829373108882343333350, −0.818285297074716609665632360703, 2.09688706439498187574173764700, 3.92155562759260510406331554334, 5.07842998125737186379546022974, 6.76172929343104872484960724361, 7.64287181234270176846247875540, 9.456967052216114373680444110037, 10.50890803345000283992344222994, 10.85572275929402021188428433258, 11.78896638926555444066672619839, 12.67159030011278152762557928054

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