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Properties

Label 2-156-156.35-c1-0-5
Degree $2$
Conductor $156$
Sign $0.519 - 0.854i$
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  + (−1.32 + 0.501i)2-s + (1.71 − 0.207i)3-s + (1.49 − 1.32i)4-s + 3.37i·5-s + (−2.16 + 1.13i)6-s + (−1.77 + 1.02i)7-s + (−1.31 + 2.50i)8-s + (2.91 − 0.713i)9-s + (−1.69 − 4.45i)10-s + (2.01 − 3.48i)11-s + (2.29 − 2.59i)12-s + (−0.274 + 3.59i)13-s + (1.83 − 2.24i)14-s + (0.699 + 5.79i)15-s + (0.479 − 3.97i)16-s + (−0.0707 + 0.0408i)17-s + ⋯
L(s)  = 1  + (−0.934 + 0.354i)2-s + (0.992 − 0.119i)3-s + (0.748 − 0.663i)4-s + 1.50i·5-s + (−0.885 + 0.464i)6-s + (−0.670 + 0.387i)7-s + (−0.464 + 0.885i)8-s + (0.971 − 0.237i)9-s + (−0.534 − 1.40i)10-s + (0.607 − 1.05i)11-s + (0.663 − 0.748i)12-s + (−0.0760 + 0.997i)13-s + (0.489 − 0.599i)14-s + (0.180 + 1.49i)15-s + (0.119 − 0.992i)16-s + (−0.0171 + 0.00991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.519 - 0.854i$
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: \((2,\ 156,\ (\ :1/2),\ 0.519 - 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.868844 + 0.488740i\)
\(L(\frac12)\) \(\approx\) \(0.868844 + 0.488740i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 - 0.501i)T \)
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

−13.45428992027845838117529556583, −11.77140992460337253057432966963, −10.89796315401752627653168810424, −9.744658553243460049886181590909, −9.101618092768699509295070814391, −7.914700335443203428919025924860, −6.83203082570824141277696391538, −6.21527550893439354367702236938, −3.49897836071258578881926099559, −2.32895927545660452693469987147, 1.42534584602849279507352721121, 3.28906335452844295008639444207, 4.69036470664089012274593080488, 6.80553377201021993011557670242, 7.964889237654869965870170876428, 8.741328503457439482580854698702, 9.666130855586566370916058654286, 10.20199450273264043310766444407, 11.97389211766318395438893181783, 12.75842589405901753536400617883

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