Properties

Label 2-171-171.112-c1-0-13
Degree $2$
Conductor $171$
Sign $0.530 + 0.847i$
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  + (0.713 − 0.259i)2-s + (0.884 − 1.48i)3-s + (−1.09 + 0.915i)4-s + (0.125 − 0.713i)5-s + (0.244 − 1.29i)6-s + (2.02 − 3.50i)7-s + (−1.29 + 2.25i)8-s + (−1.43 − 2.63i)9-s + (−0.0954 − 0.541i)10-s + 2.48·11-s + (0.397 + 2.43i)12-s + (0.692 + 3.93i)13-s + (0.533 − 3.02i)14-s + (−0.950 − 0.818i)15-s + (0.151 − 0.861i)16-s + (−1.06 + 6.06i)17-s + ⋯
L(s)  = 1  + (0.504 − 0.183i)2-s + (0.510 − 0.859i)3-s + (−0.545 + 0.457i)4-s + (0.0562 − 0.319i)5-s + (0.0998 − 0.527i)6-s + (0.764 − 1.32i)7-s + (−0.459 + 0.795i)8-s + (−0.478 − 0.878i)9-s + (−0.0301 − 0.171i)10-s + 0.748·11-s + (0.114 + 0.702i)12-s + (0.192 + 1.08i)13-s + (0.142 − 0.808i)14-s + (−0.245 − 0.211i)15-s + (0.0379 − 0.215i)16-s + (−0.259 + 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.530 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36366 - 0.755142i\)
\(L(\frac12)\) \(\approx\) \(1.36366 - 0.755142i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.884 + 1.48i)T \)
19 \( 1 + (3.54 + 2.53i)T \)
good2 \( 1 + (-0.713 + 0.259i)T + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (-0.125 + 0.713i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-2.02 + 3.50i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 2.48T + 11T^{2} \)
13 \( 1 + (-0.692 - 3.93i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.06 - 6.06i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (2.82 - 2.36i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (5.97 - 5.01i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 - 2.54T + 31T^{2} \)
37 \( 1 - 8.27T + 37T^{2} \)
41 \( 1 + (1.35 - 0.494i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.10 - 0.927i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-1.55 + 1.30i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (10.0 + 3.65i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-8.12 - 6.81i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.26 + 7.17i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (6.28 + 2.28i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-0.157 + 0.0572i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (7.43 + 6.24i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (0.362 - 2.05i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-7.16 + 12.4i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.93 + 4.14i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-9.38 + 3.41i)T + (74.3 - 62.3i)T^{2} \)
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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

−12.94354148466234778845668059717, −11.81394053013314765407059556701, −10.95186490698559185425901277762, −9.229621550674132378110873461051, −8.467898593956651067888034098135, −7.48971162021812466219600922277, −6.33945905829660953528923973377, −4.50558088158472650558776041481, −3.71988605684258376139085304796, −1.64851054875462439602630258741, 2.64176933757425952214787661871, 4.19870562125488427576812682081, 5.22443139945238279628060372501, 6.13665727239603115053435613347, 8.066260030877079866343819808941, 8.976581572347854885007176206131, 9.752554275848289464643717032798, 10.87253876143974334018683170141, 11.94537164041571940609530456086, 13.13638051491401526225187029970

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