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Properties

Label 2-156-52.43-c2-0-0
Degree $2$
Conductor $156$
Sign $-0.124 - 0.992i$
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 − 1.73i)2-s + (−1.5 − 0.866i)3-s + (−1.99 + 3.46i)4-s − 1.73i·5-s + 3.46i·6-s + (−4.95 − 8.58i)7-s + 7.99·8-s + (1.5 + 2.59i)9-s + (−2.99 + 1.73i)10-s + (−8.95 + 15.5i)11-s + (5.99 − 3.46i)12-s + (−10.4 + 7.72i)13-s + (−9.91 + 17.1i)14-s + (−1.49 + 2.59i)15-s + (−8 − 13.8i)16-s + (12.4 + 21.5i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 − 0.288i)3-s + (−0.499 + 0.866i)4-s − 0.346i·5-s + 0.577i·6-s + (−0.708 − 1.22i)7-s + 0.999·8-s + (0.166 + 0.288i)9-s + (−0.299 + 0.173i)10-s + (−0.814 + 1.41i)11-s + (0.499 − 0.288i)12-s + (−0.804 + 0.593i)13-s + (−0.708 + 1.22i)14-s + (−0.0999 + 0.173i)15-s + (−0.5 − 0.866i)16-s + (0.732 + 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $-0.124 - 0.992i$
Analytic conductor: \(4.25069\)
Root analytic conductor: \(2.06172\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1),\ -0.124 - 0.992i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0405319 + 0.0459542i\)
\(L(\frac12)\) \(\approx\) \(0.0405319 + 0.0459542i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
3 \( 1 + (1.5 + 0.866i)T \)
13 \( 1 + (10.4 - 7.72i)T \)
good5 \( 1 + 1.73iT - 25T^{2} \)
7 \( 1 + (4.95 + 8.58i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (8.95 - 15.5i)T + (-60.5 - 104. i)T^{2} \)
17 \( 1 + (-12.4 - 21.5i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (1.95 + 3.39i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (11.8 + 6.85i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (8.41 - 14.5i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + 55.8T + 961T^{2} \)
37 \( 1 + (28.3 + 16.3i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (58.1 + 33.5i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-11.8 + 6.85i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 - 23.9T + 2.20e3T^{2} \)
53 \( 1 + 1.16T + 2.80e3T^{2} \)
59 \( 1 + (-28 - 48.4i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (43.2 + 74.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (31.7 - 54.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (26.1 + 45.2i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + 126. iT - 5.32e3T^{2} \)
79 \( 1 + 34.3iT - 6.24e3T^{2} \)
83 \( 1 - 81.2T + 6.88e3T^{2} \)
89 \( 1 + (36.2 + 20.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (74.4 - 43.0i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−12.60027323289462267106045716896, −12.25319950853870657709024510096, −10.58643730102209210508636607281, −10.31059925387082280738940997944, −9.156571274984918399808873850794, −7.63502620860301644801058246596, −6.99169966467142168461178006073, −4.98622931023334668337418878214, −3.80191343608915189506214415888, −1.83829709299737051347173211989, 0.04675203268595427823241594932, 3.01963115412892966195814431213, 5.31300262739878501519126232347, 5.74876655650937212865980544862, 7.05893044502211142203515476736, 8.272572695995182144950496552876, 9.346119711871586466577402721379, 10.18316301310820286677525488035, 11.24411317627403761827804595321, 12.43492497460348772567330871146

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