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Properties

Label 2-171-171.121-c1-0-0
Degree $2$
Conductor $171$
Sign $-0.971 - 0.236i$
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  + (−0.185 + 0.321i)2-s + (−0.894 + 1.48i)3-s + (0.931 + 1.61i)4-s − 3.55·5-s + (−0.310 − 0.562i)6-s + (−0.124 − 0.216i)7-s − 1.43·8-s + (−1.39 − 2.65i)9-s + (0.659 − 1.14i)10-s + (−0.815 − 1.41i)11-s + (−3.22 − 0.0623i)12-s + (0.662 + 1.14i)13-s + 0.0926·14-s + (3.18 − 5.27i)15-s + (−1.59 + 2.76i)16-s + (3.72 + 6.46i)17-s + ⋯
L(s)  = 1  + (−0.131 + 0.227i)2-s + (−0.516 + 0.856i)3-s + (0.465 + 0.806i)4-s − 1.58·5-s + (−0.126 − 0.229i)6-s + (−0.0471 − 0.0817i)7-s − 0.506·8-s + (−0.466 − 0.884i)9-s + (0.208 − 0.361i)10-s + (−0.245 − 0.426i)11-s + (−0.931 − 0.0180i)12-s + (0.183 + 0.318i)13-s + 0.0247·14-s + (0.821 − 1.36i)15-s + (−0.399 + 0.691i)16-s + (0.904 + 1.56i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.236i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.971 - 0.236i$
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: \((2,\ 171,\ (\ :1/2),\ -0.971 - 0.236i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0667567 + 0.556592i\)
\(L(\frac12)\) \(\approx\) \(0.0667567 + 0.556592i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.894 - 1.48i)T \)
19 \( 1 + (4.07 - 1.54i)T \)
good2 \( 1 + (0.185 - 0.321i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 3.55T + 5T^{2} \)
7 \( 1 + (0.124 + 0.216i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.815 + 1.41i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.662 - 1.14i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.72 - 6.46i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.24 - 3.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.12T + 29T^{2} \)
31 \( 1 + (4.32 - 7.49i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.10T + 37T^{2} \)
41 \( 1 - 5.54T + 41T^{2} \)
43 \( 1 + (-5.02 + 8.69i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.36T + 47T^{2} \)
53 \( 1 + (-0.254 + 0.440i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 - 4.14T + 61T^{2} \)
67 \( 1 + (-0.399 - 0.692i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.60 - 9.70i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.84 + 3.20i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.92 - 8.53i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.185 - 0.320i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.01 + 6.94i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.21 - 5.56i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.73332521688780339062390574200, −12.13202022809577139371576750970, −11.17992413211721713612210388280, −10.61836573188522546898992865798, −8.903939025951671251649943896990, −8.125614638120550194902513268910, −7.10544402892709113533466744219, −5.77345663818748392520885061981, −4.02357147971554509162260395969, −3.50093918575044161238629048013, 0.56911206195925288791616412090, 2.67702049667072121933532786051, 4.66495175773052982560229352292, 5.96140750836344195142616118786, 7.22684239125390928265559356646, 7.79612172854764699719368373944, 9.290843747254566778682023556527, 10.79932563937050731506622538226, 11.29827623444704876377380193652, 12.14105786130230347891886656431

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