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Properties

Label 2-156-13.5-c2-0-1
Degree $2$
Conductor $156$
Sign $0.914 - 0.404i$
Analytic cond. $4.25069$
Root an. cond. $2.06172$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·3-s + (−0.848 + 0.848i)5-s + (5.42 + 5.42i)7-s + 2.99·9-s + (3.83 + 3.83i)11-s + (12.9 − 1.59i)13-s + (−1.46 + 1.46i)15-s − 2.95i·17-s + (−0.754 + 0.754i)19-s + (9.40 + 9.40i)21-s − 19.1i·23-s + 23.5i·25-s + 5.19·27-s − 34.3·29-s + (14.8 − 14.8i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + (−0.169 + 0.169i)5-s + (0.775 + 0.775i)7-s + 0.333·9-s + (0.348 + 0.348i)11-s + (0.992 − 0.122i)13-s + (−0.0979 + 0.0979i)15-s − 0.173i·17-s + (−0.0397 + 0.0397i)19-s + (0.447 + 0.447i)21-s − 0.831i·23-s + 0.942i·25-s + 0.192·27-s − 1.18·29-s + (0.477 − 0.477i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.914 - 0.404i$
Analytic conductor: \(4.25069\)
Root analytic conductor: \(2.06172\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1),\ 0.914 - 0.404i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.81874 + 0.384763i\)
\(L(\frac12)\) \(\approx\) \(1.81874 + 0.384763i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 1.73T \)
13 \( 1 + (-12.9 + 1.59i)T \)
good5 \( 1 + (0.848 - 0.848i)T - 25iT^{2} \)
7 \( 1 + (-5.42 - 5.42i)T + 49iT^{2} \)
11 \( 1 + (-3.83 - 3.83i)T + 121iT^{2} \)
17 \( 1 + 2.95iT - 289T^{2} \)
19 \( 1 + (0.754 - 0.754i)T - 361iT^{2} \)
23 \( 1 + 19.1iT - 529T^{2} \)
29 \( 1 + 34.3T + 841T^{2} \)
31 \( 1 + (-14.8 + 14.8i)T - 961iT^{2} \)
37 \( 1 + (-2.18 - 2.18i)T + 1.36e3iT^{2} \)
41 \( 1 + (12.9 - 12.9i)T - 1.68e3iT^{2} \)
43 \( 1 + 22.0iT - 1.84e3T^{2} \)
47 \( 1 + (52.5 + 52.5i)T + 2.20e3iT^{2} \)
53 \( 1 + 101.T + 2.80e3T^{2} \)
59 \( 1 + (32.0 + 32.0i)T + 3.48e3iT^{2} \)
61 \( 1 + 43.7T + 3.72e3T^{2} \)
67 \( 1 + (-12.5 + 12.5i)T - 4.48e3iT^{2} \)
71 \( 1 + (-41.2 + 41.2i)T - 5.04e3iT^{2} \)
73 \( 1 + (-26.4 - 26.4i)T + 5.32e3iT^{2} \)
79 \( 1 + 27.4T + 6.24e3T^{2} \)
83 \( 1 + (-96.6 + 96.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (-98.1 - 98.1i)T + 7.92e3iT^{2} \)
97 \( 1 + (26.1 - 26.1i)T - 9.40e3iT^{2} \)
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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

−12.81881661445438478408023226271, −11.71434751655851616232543072959, −10.89119054795782602938878867050, −9.522102683585003077007313687381, −8.609675292523212397934507099455, −7.73070210027502395640880943157, −6.36360226574454310302948487208, −4.95498992024679537797426143501, −3.50369228236151632686274322922, −1.87490308967734872209532679582, 1.43956879458254923958440913599, 3.49495439572271045802482207057, 4.60430464458041502922492186939, 6.24454258447296375648918593442, 7.60909445086515288776064223964, 8.390582215667555441432990509074, 9.462355273078906518723263599575, 10.74308668961063725459932178815, 11.48386012605369603596361163865, 12.79401627727492986060217676249

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