Properties

Label 2-177-59.58-c2-0-18
Degree $2$
Conductor $177$
Sign $-0.883 + 0.468i$
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  − 3.38i·2-s + 1.73·3-s − 7.45·4-s + 9.27·5-s − 5.86i·6-s − 9.96·7-s + 11.7i·8-s + 2.99·9-s − 31.3i·10-s − 20.5i·11-s − 12.9·12-s − 4.19i·13-s + 33.7i·14-s + 16.0·15-s + 9.81·16-s + 0.362·17-s + ⋯
L(s)  = 1  − 1.69i·2-s + 0.577·3-s − 1.86·4-s + 1.85·5-s − 0.977i·6-s − 1.42·7-s + 1.46i·8-s + 0.333·9-s − 3.13i·10-s − 1.86i·11-s − 1.07·12-s − 0.322i·13-s + 2.40i·14-s + 1.07·15-s + 0.613·16-s + 0.0213·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.883 + 0.468i$
Analytic conductor: \(4.82290\)
Root analytic conductor: \(2.19611\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1),\ -0.883 + 0.468i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.451992 - 1.81857i\)
\(L(\frac12)\) \(\approx\) \(0.451992 - 1.81857i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
59 \( 1 + (52.1 - 27.6i)T \)
good2 \( 1 + 3.38iT - 4T^{2} \)
5 \( 1 - 9.27T + 25T^{2} \)
7 \( 1 + 9.96T + 49T^{2} \)
11 \( 1 + 20.5iT - 121T^{2} \)
13 \( 1 + 4.19iT - 169T^{2} \)
17 \( 1 - 0.362T + 289T^{2} \)
19 \( 1 - 11.7T + 361T^{2} \)
23 \( 1 - 21.5iT - 529T^{2} \)
29 \( 1 - 18.1T + 841T^{2} \)
31 \( 1 - 42.6iT - 961T^{2} \)
37 \( 1 - 46.4iT - 1.36e3T^{2} \)
41 \( 1 - 16.1T + 1.68e3T^{2} \)
43 \( 1 + 30.4iT - 1.84e3T^{2} \)
47 \( 1 - 5.06iT - 2.20e3T^{2} \)
53 \( 1 - 36.6T + 2.80e3T^{2} \)
61 \( 1 - 32.0iT - 3.72e3T^{2} \)
67 \( 1 + 81.4iT - 4.48e3T^{2} \)
71 \( 1 - 9.92T + 5.04e3T^{2} \)
73 \( 1 + 43.0iT - 5.32e3T^{2} \)
79 \( 1 + 104.T + 6.24e3T^{2} \)
83 \( 1 - 118. iT - 6.88e3T^{2} \)
89 \( 1 + 48.1iT - 7.92e3T^{2} \)
97 \( 1 - 125. iT - 9.40e3T^{2} \)
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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

−12.15693341774228129871891346318, −10.75963866987479090130380583209, −10.08972745617601154285765049841, −9.373011990481703529999575623787, −8.708891535203694430986752784963, −6.48427639662653125632916617827, −5.42379443946664896967316402205, −3.32434198518084417775043741757, −2.81542931313248654465562691800, −1.18882402833933497018196481866, 2.38060865724437058772573282141, 4.52693316819370695991560517043, 5.81103103328579557391759849786, 6.61159568235786066137217238914, 7.34756788102180287025338220591, 8.938174794037586592912991653179, 9.641689695312551378300347306169, 10.03428789571720483096010558490, 12.68716178128707794144274675266, 13.13414841382872389714210621125

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