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Properties

Label 2-171-171.106-c1-0-13
Degree $2$
Conductor $171$
Sign $0.983 - 0.178i$
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  + (0.847 + 1.46i)2-s + (0.0521 − 1.73i)3-s + (−0.435 + 0.753i)4-s + 0.0882·5-s + (2.58 − 1.39i)6-s + (1.84 − 3.19i)7-s + 1.91·8-s + (−2.99 − 0.180i)9-s + (0.0747 + 0.129i)10-s + (−1.97 + 3.42i)11-s + (1.28 + 0.792i)12-s + (−2.03 + 3.51i)13-s + 6.25·14-s + (0.00459 − 0.152i)15-s + (2.49 + 4.31i)16-s + (−0.586 + 1.01i)17-s + ⋯
L(s)  = 1  + (0.599 + 1.03i)2-s + (0.0300 − 0.999i)3-s + (−0.217 + 0.376i)4-s + 0.0394·5-s + (1.05 − 0.567i)6-s + (0.698 − 1.20i)7-s + 0.676·8-s + (−0.998 − 0.0601i)9-s + (0.0236 + 0.0409i)10-s + (−0.596 + 1.03i)11-s + (0.370 + 0.228i)12-s + (−0.563 + 0.975i)13-s + 1.67·14-s + (0.00118 − 0.0394i)15-s + (0.622 + 1.07i)16-s + (−0.142 + 0.246i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.983 - 0.178i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.983 - 0.178i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62098 + 0.145853i\)
\(L(\frac12)\) \(\approx\) \(1.62098 + 0.145853i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0521 + 1.73i)T \)
19 \( 1 + (-3.26 + 2.89i)T \)
good2 \( 1 + (-0.847 - 1.46i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 0.0882T + 5T^{2} \)
7 \( 1 + (-1.84 + 3.19i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.97 - 3.42i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.03 - 3.51i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.586 - 1.01i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.91 - 3.31i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.56T + 29T^{2} \)
31 \( 1 + (4.14 + 7.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.88T + 37T^{2} \)
41 \( 1 + 4.66T + 41T^{2} \)
43 \( 1 + (-4.12 - 7.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.43T + 47T^{2} \)
53 \( 1 + (3.62 + 6.27i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.01T + 59T^{2} \)
61 \( 1 - 3.22T + 61T^{2} \)
67 \( 1 + (-1.45 + 2.51i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.36 - 7.56i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.43 + 5.95i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.65 - 4.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.34 + 5.80i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.41 - 7.64i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.894 + 1.55i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.24184006628563049377051346604, −12.01021720739460675153985674686, −10.98726064933599391406708700171, −9.741081658196884692165217005633, −7.988889774649226140055968846472, −7.37706156951767763780518644243, −6.71841107556673980313399045977, −5.32702391894132476551437879527, −4.27473517148119557203375674210, −1.82897085539709848336190663786, 2.48281781321006841407916297074, 3.46581589865515374309013830054, 5.01003944949459265852241669386, 5.60295707588436158565534183847, 7.925855685757095051574658217583, 8.794857161586055348629130287226, 10.19190570248418627799933447264, 10.76287886131075151872684071224, 11.86103168425767740904229612008, 12.30639903283845281193699316566

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