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Properties

Label 2-171-19.18-c2-0-14
Degree $2$
Conductor $171$
Sign $0.315 - 0.948i$
Analytic cond. $4.65941$
Root an. cond. $2.15856$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.60i·2-s − 8.99·4-s − 4·5-s − 5·7-s + 18.0i·8-s + 14.4i·10-s + 10·11-s + 3.60i·13-s + 18.0i·14-s + 28.9·16-s − 15·17-s + (−6 + 18.0i)19-s + 35.9·20-s − 36.0i·22-s − 35·23-s + ⋯
L(s)  = 1  − 1.80i·2-s − 2.24·4-s − 0.800·5-s − 0.714·7-s + 2.25i·8-s + 1.44i·10-s + 0.909·11-s + 0.277i·13-s + 1.28i·14-s + 1.81·16-s − 0.882·17-s + (−0.315 + 0.948i)19-s + 1.79·20-s − 1.63i·22-s − 1.52·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.315 - 0.948i$
Analytic conductor: \(4.65941\)
Root analytic conductor: \(2.15856\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1),\ 0.315 - 0.948i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.117669 + 0.0848527i\)
\(L(\frac12)\) \(\approx\) \(0.117669 + 0.0848527i\)
\(L(2)\) 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 \( 1 + (6 - 18.0i)T \)
good2 \( 1 + 3.60iT - 4T^{2} \)
5 \( 1 + 4T + 25T^{2} \)
7 \( 1 + 5T + 49T^{2} \)
11 \( 1 - 10T + 121T^{2} \)
13 \( 1 - 3.60iT - 169T^{2} \)
17 \( 1 + 15T + 289T^{2} \)
23 \( 1 + 35T + 529T^{2} \)
29 \( 1 + 18.0iT - 841T^{2} \)
31 \( 1 + 36.0iT - 961T^{2} \)
37 \( 1 + 21.6iT - 1.36e3T^{2} \)
41 \( 1 + 36.0iT - 1.68e3T^{2} \)
43 \( 1 + 20T + 1.84e3T^{2} \)
47 \( 1 + 10T + 2.20e3T^{2} \)
53 \( 1 - 75.7iT - 2.80e3T^{2} \)
59 \( 1 + 18.0iT - 3.48e3T^{2} \)
61 \( 1 + 40T + 3.72e3T^{2} \)
67 \( 1 - 39.6iT - 4.48e3T^{2} \)
71 \( 1 + 108. iT - 5.04e3T^{2} \)
73 \( 1 - 105T + 5.32e3T^{2} \)
79 \( 1 + 36.0iT - 6.24e3T^{2} \)
83 \( 1 - 40T + 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 - 122. iT - 9.40e3T^{2} \)
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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

−11.84960236131811649485396635033, −10.91677493390806239101244463563, −9.876854004928482671919582741511, −9.118088575752681824176868894902, −7.938642796243191723131636863330, −6.20960503546652108281345506545, −4.23796266244374398250605457553, −3.68771314640784669052715457123, −2.04984020821034493401802594745, −0.087293086882517839292025294900, 3.73080687906166294458054687854, 4.83281403600490277107489784171, 6.31264430671184852908176828513, 6.88578829635368436837022156261, 8.055517691702938353805255156018, 8.862464216143561582021478303180, 9.875307009138764980751996367538, 11.46411601605849149334951535598, 12.66371439073440530966357064368, 13.60524041461021130457958073938

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