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Properties

Label 2-156-3.2-c2-0-1
Degree $2$
Conductor $156$
Sign $-0.540 - 0.841i$
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  + (1.62 + 2.52i)3-s + 7.04i·5-s − 8.96·7-s + (−3.74 + 8.18i)9-s − 10.2i·11-s − 3.60·13-s + (−17.7 + 11.4i)15-s + 0.391i·17-s + 37.8·19-s + (−14.5 − 22.6i)21-s + 34.6i·23-s − 24.6·25-s + (−26.7 + 3.83i)27-s + 42.9i·29-s + 48.0·31-s + ⋯
L(s)  = 1  + (0.540 + 0.841i)3-s + 1.40i·5-s − 1.28·7-s + (−0.415 + 0.909i)9-s − 0.933i·11-s − 0.277·13-s + (−1.18 + 0.761i)15-s + 0.0230i·17-s + 1.99·19-s + (−0.692 − 1.07i)21-s + 1.50i·23-s − 0.986·25-s + (−0.989 + 0.141i)27-s + 1.48i·29-s + 1.55·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $-0.540 - 0.841i$
Analytic conductor: \(4.25069\)
Root analytic conductor: \(2.06172\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1),\ -0.540 - 0.841i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.659035 + 1.20676i\)
\(L(\frac12)\) \(\approx\) \(0.659035 + 1.20676i\)
\(L(2)\) 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 \( 1 + (-1.62 - 2.52i)T \)
13 \( 1 + 3.60T \)
good5 \( 1 - 7.04iT - 25T^{2} \)
7 \( 1 + 8.96T + 49T^{2} \)
11 \( 1 + 10.2iT - 121T^{2} \)
17 \( 1 - 0.391iT - 289T^{2} \)
19 \( 1 - 37.8T + 361T^{2} \)
23 \( 1 - 34.6iT - 529T^{2} \)
29 \( 1 - 42.9iT - 841T^{2} \)
31 \( 1 - 48.0T + 961T^{2} \)
37 \( 1 - 4.91T + 1.36e3T^{2} \)
41 \( 1 - 12.3iT - 1.68e3T^{2} \)
43 \( 1 - 20.9T + 1.84e3T^{2} \)
47 \( 1 + 30.8iT - 2.20e3T^{2} \)
53 \( 1 + 45.0iT - 2.80e3T^{2} \)
59 \( 1 + 62.3iT - 3.48e3T^{2} \)
61 \( 1 + 8.23T + 3.72e3T^{2} \)
67 \( 1 + 54.4T + 4.48e3T^{2} \)
71 \( 1 + 62.9iT - 5.04e3T^{2} \)
73 \( 1 - 9.45T + 5.32e3T^{2} \)
79 \( 1 - 73.0T + 6.24e3T^{2} \)
83 \( 1 + 108. iT - 6.88e3T^{2} \)
89 \( 1 - 114. iT - 7.92e3T^{2} \)
97 \( 1 + 33.0T + 9.40e3T^{2} \)
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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

−13.47305765838258178173945040521, −11.79612697287059075576911430853, −10.84695209570004636270018756610, −9.937910540502080112448146178637, −9.294177191924846940925806464016, −7.78008944669671407527294393621, −6.69449477582256545374545355030, −5.44428468153100409279768433573, −3.37838105878168329045579766357, −3.06983433246636845102464186297, 0.844000125042058709580889001960, 2.72990478804130188086989829347, 4.43443355030473631902763192385, 5.94682841351535659576978330941, 7.14282317305856635343980508458, 8.197703862760073050996465998443, 9.317744492734610376852026050648, 9.857488549725112017242297540249, 12.00769704322791378201392969679, 12.38148259531879475638878991390

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