Properties

Label 2-171-171.121-c1-0-10
Degree $2$
Conductor $171$
Sign $0.564 + 0.825i$
Analytic cond. $1.36544$
Root an. cond. $1.16852$
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  + (0.5 − 0.866i)2-s + (−1.5 − 0.866i)3-s + (0.500 + 0.866i)4-s + 3·5-s + (−1.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 3·8-s + (1.5 + 2.59i)9-s + (1.5 − 2.59i)10-s + (−2.5 − 4.33i)11-s − 1.73i·12-s + (−1 − 1.73i)13-s − 0.999·14-s + (−4.5 − 2.59i)15-s + (0.500 − 0.866i)16-s + (2.5 + 4.33i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.866 − 0.499i)3-s + (0.250 + 0.433i)4-s + 1.34·5-s + (−0.612 + 0.353i)6-s + (−0.188 − 0.327i)7-s + 1.06·8-s + (0.5 + 0.866i)9-s + (0.474 − 0.821i)10-s + (−0.753 − 1.30i)11-s − 0.499i·12-s + (−0.277 − 0.480i)13-s − 0.267·14-s + (−1.16 − 0.670i)15-s + (0.125 − 0.216i)16-s + (0.606 + 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.564 + 0.825i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.564 + 0.825i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20037 - 0.633002i\)
\(L(\frac12)\) \(\approx\) \(1.20037 - 0.633002i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 0.866i)T \)
19 \( 1 + (4 + 1.73i)T \)
good2 \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 3T + 5T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 5T + 59T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.5 + 2.59i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.88175615714191337441196109689, −11.61409407938119411404578755455, −10.67139776607494661805316815598, −10.18966887567098193450825032113, −8.424272868385385424520672541082, −7.21270551016907254918314328735, −6.03159128952584862550636425224, −5.15664700952504823783882866985, −3.23968468519268185190959903964, −1.70622857617598332888309681540, 2.10251270516430367850184925394, 4.72933796398282718705698572119, 5.36405632162187088604720244345, 6.38574148858297561264300594440, 7.16674136269042478493767604865, 9.207658597561480464347075941016, 10.13868505007393307386861931920, 10.55129676400131601881591675975, 12.04629196587565280467183349285, 12.94557418304685453704665704486

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