Properties

Label 2-171-171.128-c1-0-6
Degree $2$
Conductor $171$
Sign $0.863 + 0.504i$
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  + (0.114 − 0.651i)2-s + (−1.71 + 0.263i)3-s + (1.46 + 0.534i)4-s + (−2.11 − 0.373i)5-s + (−0.0252 + 1.14i)6-s + 4.63·7-s + (1.17 − 2.04i)8-s + (2.86 − 0.901i)9-s + (−0.486 + 1.33i)10-s + (2.06 − 1.19i)11-s + (−2.65 − 0.528i)12-s + (−1.12 − 1.34i)13-s + (0.532 − 3.02i)14-s + (3.72 + 0.0819i)15-s + (1.19 + 1.00i)16-s + (1.52 + 4.20i)17-s + ⋯
L(s)  = 1  + (0.0812 − 0.460i)2-s + (−0.988 + 0.151i)3-s + (0.733 + 0.267i)4-s + (−0.946 − 0.166i)5-s + (−0.0102 + 0.467i)6-s + 1.75·7-s + (0.416 − 0.721i)8-s + (0.953 − 0.300i)9-s + (−0.153 + 0.422i)10-s + (0.623 − 0.360i)11-s + (−0.765 − 0.152i)12-s + (−0.312 − 0.371i)13-s + (0.142 − 0.807i)14-s + (0.961 + 0.0211i)15-s + (0.299 + 0.251i)16-s + (0.371 + 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.863 + 0.504i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.863 + 0.504i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07030 - 0.290101i\)
\(L(\frac12)\) \(\approx\) \(1.07030 - 0.290101i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 - 0.263i)T \)
19 \( 1 + (3.09 + 3.06i)T \)
good2 \( 1 + (-0.114 + 0.651i)T + (-1.87 - 0.684i)T^{2} \)
5 \( 1 + (2.11 + 0.373i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 - 4.63T + 7T^{2} \)
11 \( 1 + (-2.06 + 1.19i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.12 + 1.34i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.52 - 4.20i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (1.97 - 5.41i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-5.01 + 4.21i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (2.62 + 1.51i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.49iT - 37T^{2} \)
41 \( 1 + (0.835 - 0.303i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (10.1 - 3.70i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (5.36 + 6.39i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (5.26 - 4.41i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-4.79 - 4.02i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.265 - 1.50i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (2.99 - 0.528i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (7.57 + 6.35i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-1.89 + 10.7i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (1.38 - 1.65i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + 2.14iT - 83T^{2} \)
89 \( 1 + (-2.28 - 12.9i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-1.81 - 0.320i)T + (91.1 + 33.1i)T^{2} \)
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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

−12.04679498391435585537061773559, −11.71265014033429604925953344997, −11.07937412291123305841862589557, −10.17177468881053747347773087591, −8.293965002916237590108041051429, −7.57746381713376851658206697421, −6.27374529508701771155505340425, −4.81776647370716900168617147468, −3.82210989493218153040278159122, −1.53406253979824774314505258640, 1.74350726322758619113223221039, 4.39475862274283686694706607018, 5.25456588700320672243738752871, 6.65817711599451043952167366739, 7.44753368803355224974458805527, 8.312368326216996710064806260983, 10.24824013117133170003981035303, 11.18474771426802273307004292330, 11.69605736526070812314037241103, 12.34646081955587124444670679238

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