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Properties

Label 2-171-19.11-c3-0-2
Degree $2$
Conductor $171$
Sign $-0.288 + 0.957i$
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  + (−1.45 + 2.52i)2-s + (−0.241 − 0.419i)4-s + (−10.1 + 17.4i)5-s − 1.64·7-s − 21.8·8-s + (−29.4 − 50.9i)10-s + 8.92·11-s + (19.6 + 34.0i)13-s + (2.39 − 4.14i)14-s + (33.8 − 58.5i)16-s + (−3.95 + 6.84i)17-s + (−42.6 − 70.9i)19-s + 9.77·20-s + (−12.9 + 22.5i)22-s + (55.2 + 95.7i)23-s + ⋯
L(s)  = 1  + (−0.514 + 0.891i)2-s + (−0.0302 − 0.0523i)4-s + (−0.903 + 1.56i)5-s − 0.0887·7-s − 0.967·8-s + (−0.930 − 1.61i)10-s + 0.244·11-s + (0.419 + 0.726i)13-s + (0.0456 − 0.0791i)14-s + (0.528 − 0.915i)16-s + (−0.0563 + 0.0976i)17-s + (−0.515 − 0.856i)19-s + 0.109·20-s + (−0.125 + 0.218i)22-s + (0.501 + 0.868i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 + 0.957i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.288 + 0.957i$
Analytic conductor: \(10.0893\)
Root analytic conductor: \(3.17637\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :3/2),\ -0.288 + 0.957i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.298499 - 0.401795i\)
\(L(\frac12)\) \(\approx\) \(0.298499 - 0.401795i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (42.6 + 70.9i)T \)
good2 \( 1 + (1.45 - 2.52i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (10.1 - 17.4i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + 1.64T + 343T^{2} \)
11 \( 1 - 8.92T + 1.33e3T^{2} \)
13 \( 1 + (-19.6 - 34.0i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (3.95 - 6.84i)T + (-2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (-55.2 - 95.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (71.8 + 124. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 57.1T + 2.97e4T^{2} \)
37 \( 1 - 297.T + 5.06e4T^{2} \)
41 \( 1 + (-112. + 194. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (163. - 283. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (56.2 + 97.4i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (305. + 529. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (368. - 638. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-420. - 728. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (53.2 + 92.1i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (386. - 669. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (369. - 639. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (257. - 446. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 618.T + 5.71e5T^{2} \)
89 \( 1 + (62.3 + 107. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (349. - 606. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.06103467785953648364178936554, −11.48584525053089852312857577670, −11.33539545878912318605609809039, −9.835930524457245862108419925327, −8.679103589521430418852986488855, −7.58986024081959069744637722648, −6.92393442532990976664271374508, −6.11495964908182613753303185922, −3.98710080779863320291732249302, −2.79511722399199797198566569524, 0.28071650346212535961259490371, 1.46665916810169550787583710691, 3.41519181344561502067669721455, 4.75602303906750676462328901969, 6.08638993667591146257228610411, 7.890643929019666148643131047081, 8.712622945735404713410287881832, 9.507259498961522294142052683874, 10.73153283698504006761161849856, 11.55137135701904150733923367774

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