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Properties

Label 2-156-4.3-c2-0-12
Degree $2$
Conductor $156$
Sign $0.976 - 0.216i$
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  + (−1.25 + 1.56i)2-s + 1.73i·3-s + (−0.867 − 3.90i)4-s + 6.81·5-s + (−2.70 − 2.16i)6-s − 11.2i·7-s + (7.17 + 3.53i)8-s − 2.99·9-s + (−8.52 + 10.6i)10-s − 21.6i·11-s + (6.76 − 1.50i)12-s + 3.60·13-s + (17.5 + 14.0i)14-s + 11.7i·15-s + (−14.4 + 6.77i)16-s + 12.4·17-s + ⋯
L(s)  = 1  + (−0.625 + 0.780i)2-s + 0.577i·3-s + (−0.216 − 0.976i)4-s + 1.36·5-s + (−0.450 − 0.361i)6-s − 1.60i·7-s + (0.897 + 0.441i)8-s − 0.333·9-s + (−0.852 + 1.06i)10-s − 1.97i·11-s + (0.563 − 0.125i)12-s + 0.277·13-s + (1.25 + 1.00i)14-s + 0.786i·15-s + (−0.905 + 0.423i)16-s + 0.733·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.976 - 0.216i$
Analytic conductor: \(4.25069\)
Root analytic conductor: \(2.06172\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1),\ 0.976 - 0.216i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.26570 + 0.138941i\)
\(L(\frac12)\) \(\approx\) \(1.26570 + 0.138941i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.25 - 1.56i)T \)
3 \( 1 - 1.73iT \)
13 \( 1 - 3.60T \)
good5 \( 1 - 6.81T + 25T^{2} \)
7 \( 1 + 11.2iT - 49T^{2} \)
11 \( 1 + 21.6iT - 121T^{2} \)
17 \( 1 - 12.4T + 289T^{2} \)
19 \( 1 - 18.7iT - 361T^{2} \)
23 \( 1 - 21.4iT - 529T^{2} \)
29 \( 1 + 15.8T + 841T^{2} \)
31 \( 1 - 8.39iT - 961T^{2} \)
37 \( 1 + 24.5T + 1.36e3T^{2} \)
41 \( 1 - 55.7T + 1.68e3T^{2} \)
43 \( 1 + 0.733iT - 1.84e3T^{2} \)
47 \( 1 + 16.3iT - 2.20e3T^{2} \)
53 \( 1 - 57.0T + 2.80e3T^{2} \)
59 \( 1 - 14.3iT - 3.48e3T^{2} \)
61 \( 1 + 93.9T + 3.72e3T^{2} \)
67 \( 1 - 13.0iT - 4.48e3T^{2} \)
71 \( 1 - 115. iT - 5.04e3T^{2} \)
73 \( 1 - 35.5T + 5.32e3T^{2} \)
79 \( 1 - 48.5iT - 6.24e3T^{2} \)
83 \( 1 - 31.4iT - 6.88e3T^{2} \)
89 \( 1 - 56.9T + 7.92e3T^{2} \)
97 \( 1 - 15.6T + 9.40e3T^{2} \)
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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

−13.37180597945226118773712537146, −11.13929425271435975708674554928, −10.42661490510899510546128768151, −9.748443817160080795641433407986, −8.678556743852307046035209241371, −7.53511253415079983129369485825, −6.12932617226111538472958798905, −5.48643025658485020208457434012, −3.70342313285853531033144086055, −1.11727551969279466720091831342, 1.86055894977200109524018281550, 2.55463490127593652486781815252, 4.95942744240789810044561188207, 6.27527258350852673779267167251, 7.55575435408364390040430131935, 8.987090818313818699917886591526, 9.459405770369103294956662067548, 10.49937628700792402072121287154, 11.90292625456068785367292708731, 12.53253707565114594678515848527

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