Properties

Label 2-156-156.83-c2-0-3
Degree $2$
Conductor $156$
Sign $0.0458 - 0.998i$
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  + (−1.41 + 1.41i)2-s + (−2.82 − i)3-s − 4.00i·4-s + (−5.65 − 5.65i)5-s + (5.41 − 2.58i)6-s + (−3 + 3i)7-s + (5.65 + 5.65i)8-s + (7.00 + 5.65i)9-s + 16.0·10-s + (2.82 + 2.82i)11-s + (−4.00 + 11.3i)12-s + 13i·13-s − 8.48i·14-s + (10.3 + 21.6i)15-s − 16.0·16-s + 28.2·17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.942 − 0.333i)3-s − 1.00i·4-s + (−1.13 − 1.13i)5-s + (0.902 − 0.430i)6-s + (−0.428 + 0.428i)7-s + (0.707 + 0.707i)8-s + (0.777 + 0.628i)9-s + 1.60·10-s + (0.257 + 0.257i)11-s + (−0.333 + 0.942i)12-s + i·13-s − 0.606i·14-s + (0.689 + 1.44i)15-s − 1.00·16-s + 1.66·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0458 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0458 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.0458 - 0.998i$
Analytic conductor: \(4.25069\)
Root analytic conductor: \(2.06172\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1),\ 0.0458 - 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.295609 + 0.282361i\)
\(L(\frac12)\) \(\approx\) \(0.295609 + 0.282361i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 1.41i)T \)
3 \( 1 + (2.82 + i)T \)
13 \( 1 - 13iT \)
good5 \( 1 + (5.65 + 5.65i)T + 25iT^{2} \)
7 \( 1 + (3 - 3i)T - 49iT^{2} \)
11 \( 1 + (-2.82 - 2.82i)T + 121iT^{2} \)
17 \( 1 - 28.2T + 289T^{2} \)
19 \( 1 + (17 + 17i)T + 361iT^{2} \)
23 \( 1 - 16.9iT - 529T^{2} \)
29 \( 1 - 22.6iT - 841T^{2} \)
31 \( 1 + (-21 - 21i)T + 961iT^{2} \)
37 \( 1 + (-21 - 21i)T + 1.36e3iT^{2} \)
41 \( 1 + (39.5 + 39.5i)T + 1.68e3iT^{2} \)
43 \( 1 - 14T + 1.84e3T^{2} \)
47 \( 1 + (-31.1 - 31.1i)T + 2.20e3iT^{2} \)
53 \( 1 - 50.9iT - 2.80e3T^{2} \)
59 \( 1 + (36.7 + 36.7i)T + 3.48e3iT^{2} \)
61 \( 1 - 16T + 3.72e3T^{2} \)
67 \( 1 + (-1 - i)T + 4.48e3iT^{2} \)
71 \( 1 + (-25.4 + 25.4i)T - 5.04e3iT^{2} \)
73 \( 1 + (7 + 7i)T + 5.32e3iT^{2} \)
79 \( 1 - 106iT - 6.24e3T^{2} \)
83 \( 1 + (98.9 - 98.9i)T - 6.88e3iT^{2} \)
89 \( 1 + (79.1 - 79.1i)T - 7.92e3iT^{2} \)
97 \( 1 + (1 - i)T - 9.40e3iT^{2} \)
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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

−12.58828525399803981498179684876, −11.99633477510956106189758730948, −11.02228130855753454406010346849, −9.678923935456401128414063595259, −8.716410908939950481948091264905, −7.65829993155327357596639405698, −6.69595896653536007336268518396, −5.42989733108283070817425938259, −4.42556095737157247892236822036, −1.17195829045249922364524083605, 0.44750124281330580025297804405, 3.23081506209369708197313866668, 4.09432784918889118625876335524, 6.16860934653276947360871780367, 7.33617628795450510493714192294, 8.186035130911094083494259317902, 10.10710303323321321104000398565, 10.30371468939949386171741042690, 11.38161615964525096349278293722, 12.04429828766553223392159963775

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