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Properties

Label 2-171-171.56-c3-0-18
Degree $2$
Conductor $171$
Sign $-0.417 - 0.908i$
Analytic cond. $10.0893$
Root an. cond. $3.17637$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.85 + 3.21i)2-s + (−5.00 − 1.40i)3-s + (−2.87 − 4.97i)4-s + (−3.62 + 2.09i)5-s + (13.7 − 13.4i)6-s + (11.3 − 19.6i)7-s − 8.36·8-s + (23.0 + 14.0i)9-s − 15.5i·10-s + (12.0 + 6.97i)11-s + (7.36 + 28.9i)12-s + (22.9 − 13.2i)13-s + (41.9 + 72.7i)14-s + (21.0 − 5.36i)15-s + (38.4 − 66.6i)16-s − 5.00i·17-s + ⋯
L(s)  = 1  + (−0.655 + 1.13i)2-s + (−0.962 − 0.270i)3-s + (−0.359 − 0.621i)4-s + (−0.324 + 0.187i)5-s + (0.938 − 0.915i)6-s + (0.611 − 1.05i)7-s − 0.369·8-s + (0.853 + 0.521i)9-s − 0.490i·10-s + (0.331 + 0.191i)11-s + (0.177 + 0.695i)12-s + (0.489 − 0.282i)13-s + (0.801 + 1.38i)14-s + (0.362 − 0.0923i)15-s + (0.601 − 1.04i)16-s − 0.0714i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.417 - 0.908i$
Analytic conductor: \(10.0893\)
Root analytic conductor: \(3.17637\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :3/2),\ -0.417 - 0.908i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.401221 + 0.626239i\)
\(L(\frac12)\) \(\approx\) \(0.401221 + 0.626239i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.00 + 1.40i)T \)
19 \( 1 + (-8.87 - 82.3i)T \)
good2 \( 1 + (1.85 - 3.21i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (3.62 - 2.09i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + (-11.3 + 19.6i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-12.0 - 6.97i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-22.9 + 13.2i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 5.00iT - 4.91e3T^{2} \)
23 \( 1 + (34.7 - 20.0i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (122. - 211. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-198. + 114. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 218. iT - 5.06e4T^{2} \)
41 \( 1 + (-102. - 177. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (232. - 402. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-476. - 274. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 434.T + 1.48e5T^{2} \)
59 \( 1 + (-11.7 - 20.4i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (253. - 439. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-549. + 317. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 443.T + 3.57e5T^{2} \)
73 \( 1 - 540.T + 3.89e5T^{2} \)
79 \( 1 + (-456. - 263. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-671. - 387. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 449.T + 7.04e5T^{2} \)
97 \( 1 + (206. + 119. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.50519981092621776242429049383, −11.47791373697738286044599851294, −10.63873740253623464430810964421, −9.516695711101080416558032208951, −7.976576571723982671644698137299, −7.47975801194023030080028231123, −6.49381524738788609359104352521, −5.45788355710852634314825726908, −3.97298987715727148887066598907, −1.08472524896884338110988669712, 0.61227964036816730667621977655, 2.17160363205513048647156337428, 3.97801671041116309697120617277, 5.38928249155330105649152699404, 6.50023873953055391476463489008, 8.324098101328895524094043503502, 9.148822644780117856227390802867, 10.18220134956857858254853044901, 11.13311598587267527175829298712, 11.88627256495531481230187532619

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