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Properties

Label 2-156-52.51-c2-0-23
Degree $2$
Conductor $156$
Sign $-0.0663 + 0.997i$
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  + (0.570 + 1.91i)2-s − 1.73i·3-s + (−3.34 + 2.18i)4-s − 7.79i·5-s + (3.32 − 0.988i)6-s − 7.84·7-s + (−6.10 − 5.17i)8-s − 2.99·9-s + (14.9 − 4.44i)10-s − 13.1·11-s + (3.78 + 5.80i)12-s + (−10.3 − 7.81i)13-s + (−4.47 − 15.0i)14-s − 13.5·15-s + (6.43 − 14.6i)16-s + 22.5·17-s + ⋯
L(s)  = 1  + (0.285 + 0.958i)2-s − 0.577i·3-s + (−0.837 + 0.546i)4-s − 1.55i·5-s + (0.553 − 0.164i)6-s − 1.12·7-s + (−0.762 − 0.646i)8-s − 0.333·9-s + (1.49 − 0.444i)10-s − 1.19·11-s + (0.315 + 0.483i)12-s + (−0.799 − 0.601i)13-s + (−0.319 − 1.07i)14-s − 0.900·15-s + (0.402 − 0.915i)16-s + 1.32·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0663 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0663 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $-0.0663 + 0.997i$
Analytic conductor: \(4.25069\)
Root analytic conductor: \(2.06172\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1),\ -0.0663 + 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.549727 - 0.587501i\)
\(L(\frac12)\) \(\approx\) \(0.549727 - 0.587501i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.570 - 1.91i)T \)
3 \( 1 + 1.73iT \)
13 \( 1 + (10.3 + 7.81i)T \)
good5 \( 1 + 7.79iT - 25T^{2} \)
7 \( 1 + 7.84T + 49T^{2} \)
11 \( 1 + 13.1T + 121T^{2} \)
17 \( 1 - 22.5T + 289T^{2} \)
19 \( 1 - 26.2T + 361T^{2} \)
23 \( 1 - 10.4iT - 529T^{2} \)
29 \( 1 - 37.2T + 841T^{2} \)
31 \( 1 + 49.6T + 961T^{2} \)
37 \( 1 + 38.6iT - 1.36e3T^{2} \)
41 \( 1 + 11.4iT - 1.68e3T^{2} \)
43 \( 1 + 67.3iT - 1.84e3T^{2} \)
47 \( 1 + 10.7T + 2.20e3T^{2} \)
53 \( 1 + 57.9T + 2.80e3T^{2} \)
59 \( 1 - 16.0T + 3.48e3T^{2} \)
61 \( 1 + 55.0T + 3.72e3T^{2} \)
67 \( 1 - 36.6T + 4.48e3T^{2} \)
71 \( 1 - 56.5T + 5.04e3T^{2} \)
73 \( 1 - 37.3iT - 5.32e3T^{2} \)
79 \( 1 + 57.2iT - 6.24e3T^{2} \)
83 \( 1 - 14.2T + 6.88e3T^{2} \)
89 \( 1 + 10.3iT - 7.92e3T^{2} \)
97 \( 1 - 30.8iT - 9.40e3T^{2} \)
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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

−12.58087605216254530276981384147, −12.23221661179730636922339536369, −9.983995380409356278574980580538, −9.166396653266621663227874774730, −7.985738767829672191830743942696, −7.31389927265098664733962507007, −5.63816279254815607728737576727, −5.16571552549704941262525920283, −3.29313266430645079940205706449, −0.45267881554513677424231941060, 2.80841390413321109974934652700, 3.32914064705837892074447584719, 5.07795276594039435415568271405, 6.35903741100628741525987686346, 7.72479108604644837623797513434, 9.610581314525159197410107906087, 9.993444345644165534986564744707, 10.82575460724673543391873035368, 11.83000314018521295826144279873, 12.85397845122892742828807453962

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