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Properties

Label 2-177-3.2-c2-0-12
Degree $2$
Conductor $177$
Sign $-0.976 + 0.217i$
Analytic cond. $4.82290$
Root an. cond. $2.19611$
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  + 2.79i·2-s + (0.651 + 2.92i)3-s − 3.81·4-s + 4.04i·5-s + (−8.18 + 1.82i)6-s + 6.95·7-s + 0.509i·8-s + (−8.15 + 3.81i)9-s − 11.3·10-s − 3.60i·11-s + (−2.48 − 11.1i)12-s + 6.03·13-s + 19.4i·14-s + (−11.8 + 2.63i)15-s − 16.6·16-s − 27.5i·17-s + ⋯
L(s)  = 1  + 1.39i·2-s + (0.217 + 0.976i)3-s − 0.954·4-s + 0.808i·5-s + (−1.36 + 0.303i)6-s + 0.993·7-s + 0.0636i·8-s + (−0.905 + 0.423i)9-s − 1.13·10-s − 0.327i·11-s + (−0.207 − 0.931i)12-s + 0.463·13-s + 1.38i·14-s + (−0.789 + 0.175i)15-s − 1.04·16-s − 1.62i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.217i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.976 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.976 + 0.217i$
Analytic conductor: \(4.82290\)
Root analytic conductor: \(2.19611\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1),\ -0.976 + 0.217i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.182300 - 1.65998i\)
\(L(\frac12)\) \(\approx\) \(0.182300 - 1.65998i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.651 - 2.92i)T \)
59 \( 1 + 7.68iT \)
good2 \( 1 - 2.79iT - 4T^{2} \)
5 \( 1 - 4.04iT - 25T^{2} \)
7 \( 1 - 6.95T + 49T^{2} \)
11 \( 1 + 3.60iT - 121T^{2} \)
13 \( 1 - 6.03T + 169T^{2} \)
17 \( 1 + 27.5iT - 289T^{2} \)
19 \( 1 - 11.5T + 361T^{2} \)
23 \( 1 - 6.63iT - 529T^{2} \)
29 \( 1 + 7.59iT - 841T^{2} \)
31 \( 1 - 40.0T + 961T^{2} \)
37 \( 1 + 53.5T + 1.36e3T^{2} \)
41 \( 1 - 33.0iT - 1.68e3T^{2} \)
43 \( 1 + 19.3T + 1.84e3T^{2} \)
47 \( 1 - 51.4iT - 2.20e3T^{2} \)
53 \( 1 - 40.9iT - 2.80e3T^{2} \)
61 \( 1 - 48.9T + 3.72e3T^{2} \)
67 \( 1 - 85.3T + 4.48e3T^{2} \)
71 \( 1 - 32.2iT - 5.04e3T^{2} \)
73 \( 1 - 71.6T + 5.32e3T^{2} \)
79 \( 1 + 85.8T + 6.24e3T^{2} \)
83 \( 1 + 131. iT - 6.88e3T^{2} \)
89 \( 1 - 96.1iT - 7.92e3T^{2} \)
97 \( 1 + 80.8T + 9.40e3T^{2} \)
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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

−13.93412855019559765774762958467, −11.57803527058354110989532413886, −11.07061955784256383632760738088, −9.788511250940223609792103981599, −8.670335317977843355696857675980, −7.83827573707144839121365773146, −6.77152634788859122937962262723, −5.49521843333468464130186124480, −4.64425939036767659079000552477, −2.93094416412700754803819381035, 1.10922538316966126789794824765, 2.02792484230151286249533864160, 3.69883813627867991391803631952, 5.12952592119713461433428358457, 6.72488924293090972201610804296, 8.212858369023795067834565123958, 8.782047456079929974864771492753, 10.19007549063417340344684111765, 11.19593446572248560819502303321, 12.05122745732307749224625300764

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