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Properties

Label 2-171-171.110-c1-0-13
Degree $2$
Conductor $171$
Sign $0.907 + 0.419i$
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  + (1.30 − 1.09i)2-s + (0.387 + 1.68i)3-s + (0.158 − 0.899i)4-s + (1.05 − 2.90i)5-s + (2.35 + 1.78i)6-s + (−0.421 − 0.730i)7-s + (0.927 + 1.60i)8-s + (−2.69 + 1.30i)9-s + (−1.80 − 4.95i)10-s + 5.25i·11-s + (1.58 − 0.0811i)12-s + (−2.06 − 5.67i)13-s + (−1.35 − 0.492i)14-s + (5.30 + 0.657i)15-s + (4.69 + 1.70i)16-s + (0.911 − 2.50i)17-s + ⋯
L(s)  = 1  + (0.924 − 0.775i)2-s + (0.223 + 0.974i)3-s + (0.0793 − 0.449i)4-s + (0.472 − 1.29i)5-s + (0.963 + 0.727i)6-s + (−0.159 − 0.275i)7-s + (0.327 + 0.567i)8-s + (−0.899 + 0.436i)9-s + (−0.569 − 1.56i)10-s + 1.58i·11-s + (0.456 − 0.0234i)12-s + (−0.572 − 1.57i)13-s + (−0.361 − 0.131i)14-s + (1.36 + 0.169i)15-s + (1.17 + 0.426i)16-s + (0.221 − 0.607i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.907 + 0.419i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.907 + 0.419i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85687 - 0.408397i\)
\(L(\frac12)\) \(\approx\) \(1.85687 - 0.408397i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.387 - 1.68i)T \)
19 \( 1 + (3.91 - 1.92i)T \)
good2 \( 1 + (-1.30 + 1.09i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (-1.05 + 2.90i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.421 + 0.730i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 5.25iT - 11T^{2} \)
13 \( 1 + (2.06 + 5.67i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.911 + 2.50i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (5.64 + 0.995i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (0.889 - 5.04i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 - 4.15iT - 31T^{2} \)
37 \( 1 + 4.72iT - 37T^{2} \)
41 \( 1 + (2.20 - 1.85i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (0.757 + 4.29i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-5.55 - 0.979i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-9.57 - 8.03i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (0.518 + 2.94i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-6.29 + 2.29i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.11 - 1.33i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-5.22 + 4.38i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.376 + 2.13i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-1.27 + 3.51i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-7.69 + 4.44i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.29 - 7.33i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-1.81 - 2.16i)T + (-16.8 + 95.5i)T^{2} \)
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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

−12.51900302612433762044258752640, −12.18672899857493034458516226954, −10.51326152878653930682840436893, −10.03485520792213490620331687076, −8.848901479149881266093388426282, −7.73487496751798895406095028563, −5.44866045564117867714965845889, −4.86114452386002843752406440360, −3.84889817338489063272967499478, −2.27731357854904212441824830815, 2.37380680424676198446337872767, 3.84940659555384118565077710424, 5.87511383702001386205943559588, 6.32788918767248900796750996106, 7.14269203568923450247201593839, 8.405616909986732949689092205811, 9.808866462075357779346346096148, 11.10540810240802168001157270310, 12.04062359313350450017431844843, 13.36990522672707916366551368063

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