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Properties

Label 2-156-4.3-c2-0-14
Degree $2$
Conductor $156$
Sign $0.641 - 0.767i$
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  + (1.88 + 0.682i)2-s + 1.73i·3-s + (3.06 + 2.56i)4-s + 5.46·5-s + (−1.18 + 3.25i)6-s − 10.4i·7-s + (4.02 + 6.91i)8-s − 2.99·9-s + (10.2 + 3.72i)10-s + 10.0i·11-s + (−4.44 + 5.31i)12-s − 3.60·13-s + (7.14 − 19.6i)14-s + 9.45i·15-s + (2.84 + 15.7i)16-s − 30.3·17-s + ⋯
L(s)  = 1  + (0.940 + 0.341i)2-s + 0.577i·3-s + (0.767 + 0.641i)4-s + 1.09·5-s + (−0.196 + 0.542i)6-s − 1.49i·7-s + (0.502 + 0.864i)8-s − 0.333·9-s + (1.02 + 0.372i)10-s + 0.916i·11-s + (−0.370 + 0.443i)12-s − 0.277·13-s + (0.510 − 1.40i)14-s + 0.630i·15-s + (0.177 + 0.984i)16-s − 1.78·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.641 - 0.767i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.641 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.641 - 0.767i$
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: \((2,\ 156,\ (\ :1),\ 0.641 - 0.767i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.50620 + 1.17191i\)
\(L(\frac12)\) \(\approx\) \(2.50620 + 1.17191i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.88 - 0.682i)T \)
3 \( 1 - 1.73iT \)
13 \( 1 + 3.60T \)
good5 \( 1 - 5.46T + 25T^{2} \)
7 \( 1 + 10.4iT - 49T^{2} \)
11 \( 1 - 10.0iT - 121T^{2} \)
17 \( 1 + 30.3T + 289T^{2} \)
19 \( 1 - 5.03iT - 361T^{2} \)
23 \( 1 + 30.6iT - 529T^{2} \)
29 \( 1 - 55.0T + 841T^{2} \)
31 \( 1 + 15.5iT - 961T^{2} \)
37 \( 1 + 61.6T + 1.36e3T^{2} \)
41 \( 1 - 12.5T + 1.68e3T^{2} \)
43 \( 1 + 27.5iT - 1.84e3T^{2} \)
47 \( 1 + 27.7iT - 2.20e3T^{2} \)
53 \( 1 - 33.5T + 2.80e3T^{2} \)
59 \( 1 - 64.1iT - 3.48e3T^{2} \)
61 \( 1 + 3.73T + 3.72e3T^{2} \)
67 \( 1 - 113. iT - 4.48e3T^{2} \)
71 \( 1 + 65.4iT - 5.04e3T^{2} \)
73 \( 1 + 6.33T + 5.32e3T^{2} \)
79 \( 1 + 13.1iT - 6.24e3T^{2} \)
83 \( 1 + 54.8iT - 6.88e3T^{2} \)
89 \( 1 - 104.T + 7.92e3T^{2} \)
97 \( 1 + 38.3T + 9.40e3T^{2} \)
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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

−13.21319629523562417534050341998, −12.03375784692968988525802118622, −10.59016417593532408316583642449, −10.21473314617648238628376159637, −8.692216446218028479055774281071, −7.14908929593769042926063452098, −6.39319547530685018237208397014, −4.87134179771598348474350128818, −4.12641648265313492682897115429, −2.30366590167319982224432571051, 1.90201273145054830627157198653, 2.90107725152903919658132824722, 5.05127668235600982050804058277, 5.94476427432415617000440396863, 6.70563951204075878311758114479, 8.564541182765517356561676363406, 9.524902294767043697414673048101, 10.88792581607315153132472552789, 11.79295676808532711142561217528, 12.64686376705972097451278682800

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