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Properties

Label 2-156-52.11-c1-0-1
Degree $2$
Conductor $156$
Sign $-0.0789 - 0.996i$
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  + (−1.15 + 0.813i)2-s + (−0.866 + 0.5i)3-s + (0.677 − 1.88i)4-s + (1.41 + 1.41i)5-s + (0.595 − 1.28i)6-s + (0.844 − 0.226i)7-s + (0.746 + 2.72i)8-s + (0.499 − 0.866i)9-s + (−2.77 − 0.484i)10-s + (−1.07 + 4.02i)11-s + (0.354 + 1.96i)12-s + (−0.495 + 3.57i)13-s + (−0.792 + 0.948i)14-s + (−1.92 − 0.516i)15-s + (−3.08 − 2.54i)16-s + (1.03 + 0.596i)17-s + ⋯
L(s)  = 1  + (−0.818 + 0.575i)2-s + (−0.499 + 0.288i)3-s + (0.338 − 0.940i)4-s + (0.630 + 0.630i)5-s + (0.243 − 0.523i)6-s + (0.319 − 0.0855i)7-s + (0.264 + 0.964i)8-s + (0.166 − 0.288i)9-s + (−0.878 − 0.153i)10-s + (−0.325 + 1.21i)11-s + (0.102 + 0.568i)12-s + (−0.137 + 0.990i)13-s + (−0.211 + 0.253i)14-s + (−0.497 − 0.133i)15-s + (−0.770 − 0.637i)16-s + (0.250 + 0.144i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0789 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0789 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $-0.0789 - 0.996i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ -0.0789 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.493436 + 0.534040i\)
\(L(\frac12)\) \(\approx\) \(0.493436 + 0.534040i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 - 0.813i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (0.495 - 3.57i)T \)
good5 \( 1 + (-1.41 - 1.41i)T + 5iT^{2} \)
7 \( 1 + (-0.844 + 0.226i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.07 - 4.02i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.03 - 0.596i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.113 - 0.424i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.80 - 4.86i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.58 + 6.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.11 + 3.11i)T - 31iT^{2} \)
37 \( 1 + (-9.94 - 2.66i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.53 + 5.72i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-2.62 + 4.55i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.03 + 6.03i)T + 47iT^{2} \)
53 \( 1 + 7.41T + 53T^{2} \)
59 \( 1 + (-1.37 + 0.367i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.725 + 1.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.7 - 3.14i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.85 + 14.3i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (8.51 - 8.51i)T - 73iT^{2} \)
79 \( 1 + 9.00iT - 79T^{2} \)
83 \( 1 + (2.23 - 2.23i)T - 83iT^{2} \)
89 \( 1 + (-14.6 - 3.92i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-5.99 + 1.60i)T + (84.0 - 48.5i)T^{2} \)
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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

−13.39827847650125207331169759652, −11.84045805240783741048957495499, −10.98056836150332688273344560224, −9.906753893856910082947437555178, −9.468156504188240170570645655573, −7.83785003480513738783513567533, −6.87494361396107228609394196419, −5.88038059505076005170242014179, −4.61011559346267291014896163568, −2.04711428212833894919376951898, 1.06473457771729636071132855063, 2.94222304484590508969270169451, 5.00745937794176778055411664386, 6.23050409677661742473711787638, 7.73995835638083200315969615690, 8.601512889375140518907455842735, 9.666935184322241634820407534820, 10.78677649152964445839917021832, 11.41355742140269295264203730556, 12.85943971156641391665482975278

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