Properties

Label 2-156-3.2-c2-0-2
Degree $2$
Conductor $156$
Sign $0.0405 - 0.999i$
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  + (−0.121 + 2.99i)3-s + 0.347i·5-s + 10.5·7-s + (−8.97 − 0.729i)9-s + 16.7i·11-s − 3.60·13-s + (−1.04 − 0.0423i)15-s + 27.6i·17-s − 22.1·19-s + (−1.28 + 31.6i)21-s − 32.7i·23-s + 24.8·25-s + (3.27 − 26.8i)27-s − 8.31i·29-s + 48.0·31-s + ⋯
L(s)  = 1  + (−0.0405 + 0.999i)3-s + 0.0695i·5-s + 1.51·7-s + (−0.996 − 0.0810i)9-s + 1.52i·11-s − 0.277·13-s + (−0.0695 − 0.00282i)15-s + 1.62i·17-s − 1.16·19-s + (−0.0612 + 1.50i)21-s − 1.42i·23-s + 0.995·25-s + (0.121 − 0.992i)27-s − 0.286i·29-s + 1.55·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0405 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0405 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.0405 - 0.999i$
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: \((2,\ 156,\ (\ :1),\ 0.0405 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.08840 + 1.04514i\)
\(L(\frac12)\) \(\approx\) \(1.08840 + 1.04514i\)
\(L(2)\) 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 \( 1 + (0.121 - 2.99i)T \)
13 \( 1 + 3.60T \)
good5 \( 1 - 0.347iT - 25T^{2} \)
7 \( 1 - 10.5T + 49T^{2} \)
11 \( 1 - 16.7iT - 121T^{2} \)
17 \( 1 - 27.6iT - 289T^{2} \)
19 \( 1 + 22.1T + 361T^{2} \)
23 \( 1 + 32.7iT - 529T^{2} \)
29 \( 1 + 8.31iT - 841T^{2} \)
31 \( 1 - 48.0T + 961T^{2} \)
37 \( 1 - 3.53T + 1.36e3T^{2} \)
41 \( 1 - 17.8iT - 1.68e3T^{2} \)
43 \( 1 - 34.1T + 1.84e3T^{2} \)
47 \( 1 + 70.4iT - 2.20e3T^{2} \)
53 \( 1 + 26.2iT - 2.80e3T^{2} \)
59 \( 1 + 54.7iT - 3.48e3T^{2} \)
61 \( 1 + 68.2T + 3.72e3T^{2} \)
67 \( 1 + 12.6T + 4.48e3T^{2} \)
71 \( 1 - 10.4iT - 5.04e3T^{2} \)
73 \( 1 + 11.4T + 5.32e3T^{2} \)
79 \( 1 - 52.1T + 6.24e3T^{2} \)
83 \( 1 + 70.6iT - 6.88e3T^{2} \)
89 \( 1 + 101. iT - 7.92e3T^{2} \)
97 \( 1 + 72.1T + 9.40e3T^{2} \)
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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

−12.77402701048244758789341709859, −11.84683738042675368227939805945, −10.66993422253347610566829555821, −10.22424185868203591903022250422, −8.752762401296609667918493526220, −8.019809030073390134029535769052, −6.42108851861256615796534172791, −4.82615435424925963463863846310, −4.30570031532049057026805771579, −2.16820436151472294340094829721, 1.06546847504229657388565316791, 2.75765068411766563809168520989, 4.83561428504815834148974472841, 5.95519946749671064679328927234, 7.31622891239393450644180032989, 8.191811197486263664753142285732, 9.026673066842119911573625178806, 10.92025468945314552798650645383, 11.43056986436886538919383687330, 12.35794247197874735246560279332

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