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Properties

Label 2-156-12.11-c1-0-18
Degree $2$
Conductor $156$
Sign $0.995 + 0.0927i$
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  + (1.11 + 0.866i)2-s + (0.586 − 1.62i)3-s + (0.500 + 1.93i)4-s − 3.82i·5-s + (2.06 − 1.31i)6-s + 3.25i·7-s + (−1.11 + 2.59i)8-s + (−2.31 − 1.91i)9-s + (3.31 − 4.27i)10-s + (3.44 + 0.321i)12-s + 13-s + (−2.82 + 3.64i)14-s + (−6.23 − 2.24i)15-s + (−3.5 + 1.93i)16-s + 3.82i·17-s + (−0.928 − 4.13i)18-s + ⋯
L(s)  = 1  + (0.790 + 0.612i)2-s + (0.338 − 0.940i)3-s + (0.250 + 0.968i)4-s − 1.71i·5-s + (0.843 − 0.536i)6-s + 1.23i·7-s + (−0.395 + 0.918i)8-s + (−0.770 − 0.637i)9-s + (1.04 − 1.35i)10-s + (0.995 + 0.0927i)12-s + 0.277·13-s + (−0.754 + 0.973i)14-s + (−1.60 − 0.579i)15-s + (−0.875 + 0.484i)16-s + 0.927i·17-s + (−0.218 − 0.975i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.995 + 0.0927i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ 0.995 + 0.0927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75820 - 0.0816774i\)
\(L(\frac12)\) \(\approx\) \(1.75820 - 0.0816774i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.11 - 0.866i)T \)
3 \( 1 + (-0.586 + 1.62i)T \)
13 \( 1 - T \)
good5 \( 1 + 3.82iT - 5T^{2} \)
7 \( 1 - 3.25iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 - 3.82iT - 17T^{2} \)
19 \( 1 - 5.29iT - 19T^{2} \)
23 \( 1 + 2.34T + 23T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 + 5.29iT - 31T^{2} \)
37 \( 1 - 4.62T + 37T^{2} \)
41 \( 1 - 0.719iT - 41T^{2} \)
43 \( 1 + 7.32iT - 43T^{2} \)
47 \( 1 + 1.17T + 47T^{2} \)
53 \( 1 + 0.719iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 5.29iT - 67T^{2} \)
71 \( 1 + 1.17T + 71T^{2} \)
73 \( 1 - 7.24T + 73T^{2} \)
79 \( 1 + 1.22iT - 79T^{2} \)
83 \( 1 - 4.69T + 83T^{2} \)
89 \( 1 - 14.5iT - 89T^{2} \)
97 \( 1 - 15.2T + 97T^{2} \)
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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

−12.91356696851178288505864338183, −12.26061510270211812766745835562, −11.73329766225640396738675509262, −9.333414580402539154643540154145, −8.369000663485634391513115533095, −7.964498364205499229963460785611, −6.10669187462974166608575654638, −5.56847037861222249784840693986, −4.01421201245683101786059422187, −2.05868639853871954857846462661, 2.77878981890764554380270789862, 3.63137557525668226292285310873, 4.84388369322217865405670124977, 6.47253845551160775631889560701, 7.40565288617832594864250603661, 9.366696825054488723033336265198, 10.36742057514727639479689525036, 10.84187856177014997023355865548, 11.54236834620042240545290466360, 13.32258018615220907837873379814

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