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Properties

Label 2-156-13.8-c2-0-2
Degree $2$
Conductor $156$
Sign $0.478 + 0.878i$
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  − 1.73·3-s + (−0.658 − 0.658i)5-s + (1.58 − 1.58i)7-s + 2.99·9-s + (15.0 − 15.0i)11-s + (−9.25 − 9.12i)13-s + (1.14 + 1.14i)15-s − 25.0i·17-s + (17.2 + 17.2i)19-s + (−2.74 + 2.74i)21-s + 3.84i·23-s − 24.1i·25-s − 5.19·27-s + 18.0·29-s + (1.56 + 1.56i)31-s + ⋯
L(s)  = 1  − 0.577·3-s + (−0.131 − 0.131i)5-s + (0.226 − 0.226i)7-s + 0.333·9-s + (1.36 − 1.36i)11-s + (−0.712 − 0.702i)13-s + (0.0760 + 0.0760i)15-s − 1.47i·17-s + (0.909 + 0.909i)19-s + (−0.130 + 0.130i)21-s + 0.167i·23-s − 0.965i·25-s − 0.192·27-s + 0.621·29-s + (0.0506 + 0.0506i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.478 + 0.878i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.478 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.478 + 0.878i$
Analytic conductor: \(4.25069\)
Root analytic conductor: \(2.06172\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1),\ 0.478 + 0.878i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.01424 - 0.602698i\)
\(L(\frac12)\) \(\approx\) \(1.01424 - 0.602698i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + 1.73T \)
13 \( 1 + (9.25 + 9.12i)T \)
good5 \( 1 + (0.658 + 0.658i)T + 25iT^{2} \)
7 \( 1 + (-1.58 + 1.58i)T - 49iT^{2} \)
11 \( 1 + (-15.0 + 15.0i)T - 121iT^{2} \)
17 \( 1 + 25.0iT - 289T^{2} \)
19 \( 1 + (-17.2 - 17.2i)T + 361iT^{2} \)
23 \( 1 - 3.84iT - 529T^{2} \)
29 \( 1 - 18.0T + 841T^{2} \)
31 \( 1 + (-1.56 - 1.56i)T + 961iT^{2} \)
37 \( 1 + (47.4 - 47.4i)T - 1.36e3iT^{2} \)
41 \( 1 + (23.2 + 23.2i)T + 1.68e3iT^{2} \)
43 \( 1 - 37.7iT - 1.84e3T^{2} \)
47 \( 1 + (-34.6 + 34.6i)T - 2.20e3iT^{2} \)
53 \( 1 - 27.0T + 2.80e3T^{2} \)
59 \( 1 + (27.0 - 27.0i)T - 3.48e3iT^{2} \)
61 \( 1 + 67.7T + 3.72e3T^{2} \)
67 \( 1 + (-10.6 - 10.6i)T + 4.48e3iT^{2} \)
71 \( 1 + (-37.3 - 37.3i)T + 5.04e3iT^{2} \)
73 \( 1 + (-51.5 + 51.5i)T - 5.32e3iT^{2} \)
79 \( 1 - 105.T + 6.24e3T^{2} \)
83 \( 1 + (89.5 + 89.5i)T + 6.88e3iT^{2} \)
89 \( 1 + (-7.20 + 7.20i)T - 7.92e3iT^{2} \)
97 \( 1 + (-5.87 - 5.87i)T + 9.40e3iT^{2} \)
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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

−12.08946959991429715984852642903, −11.80100164625319844758596442764, −10.59549460217150729972964408611, −9.554513644553715296626185260107, −8.360543156360068755083729069336, −7.15743237892358574417877454808, −5.98081492665635948162457326932, −4.82502667260433278618287203835, −3.30962342154773955384675972168, −0.885136588302033539763289389000, 1.77334465423322805867314741825, 3.97614764359033392902706789167, 5.10730603217487490354146809626, 6.59227489483133514002811491932, 7.33719370480305369221941899762, 8.945992469628517802766236487653, 9.813952135973283115757026385159, 10.99691229080541256805042895586, 12.00233166894588754982849393679, 12.49227308480117738357757588868

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