Properties

Label 2-177-3.2-c2-0-33
Degree $2$
Conductor $177$
Sign $-0.998 - 0.0602i$
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  − 1.89i·2-s + (0.180 − 2.99i)3-s + 0.403·4-s + 2.43i·5-s + (−5.67 − 0.342i)6-s − 8.99·7-s − 8.35i·8-s + (−8.93 − 1.08i)9-s + 4.62·10-s − 9.24i·11-s + (0.0729 − 1.20i)12-s − 11.6·13-s + 17.0i·14-s + (7.30 + 0.440i)15-s − 14.2·16-s − 27.6i·17-s + ⋯
L(s)  = 1  − 0.948i·2-s + (0.0602 − 0.998i)3-s + 0.100·4-s + 0.487i·5-s + (−0.946 − 0.0571i)6-s − 1.28·7-s − 1.04i·8-s + (−0.992 − 0.120i)9-s + 0.462·10-s − 0.840i·11-s + (0.00608 − 0.100i)12-s − 0.896·13-s + 1.21i·14-s + (0.486 + 0.0293i)15-s − 0.888·16-s − 1.62i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0602i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.998 - 0.0602i$
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: \((2,\ 177,\ (\ :1),\ -0.998 - 0.0602i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0363917 + 1.20695i\)
\(L(\frac12)\) \(\approx\) \(0.0363917 + 1.20695i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.180 + 2.99i)T \)
59 \( 1 - 7.68iT \)
good2 \( 1 + 1.89iT - 4T^{2} \)
5 \( 1 - 2.43iT - 25T^{2} \)
7 \( 1 + 8.99T + 49T^{2} \)
11 \( 1 + 9.24iT - 121T^{2} \)
13 \( 1 + 11.6T + 169T^{2} \)
17 \( 1 + 27.6iT - 289T^{2} \)
19 \( 1 - 29.8T + 361T^{2} \)
23 \( 1 - 29.8iT - 529T^{2} \)
29 \( 1 + 16.6iT - 841T^{2} \)
31 \( 1 + 11.5T + 961T^{2} \)
37 \( 1 - 68.3T + 1.36e3T^{2} \)
41 \( 1 + 2.67iT - 1.68e3T^{2} \)
43 \( 1 - 14.6T + 1.84e3T^{2} \)
47 \( 1 + 15.0iT - 2.20e3T^{2} \)
53 \( 1 + 67.2iT - 2.80e3T^{2} \)
61 \( 1 - 102.T + 3.72e3T^{2} \)
67 \( 1 + 55.2T + 4.48e3T^{2} \)
71 \( 1 - 123. iT - 5.04e3T^{2} \)
73 \( 1 + 81.1T + 5.32e3T^{2} \)
79 \( 1 + 45.7T + 6.24e3T^{2} \)
83 \( 1 + 144. iT - 6.88e3T^{2} \)
89 \( 1 + 8.00iT - 7.92e3T^{2} \)
97 \( 1 + 41.6T + 9.40e3T^{2} \)
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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

−11.66542908419742452010290051727, −11.49121365763809645365554891508, −9.936679302500684731617474087109, −9.316022235643213997207072699916, −7.45935570393606055539119695318, −6.89830974449278095367768051335, −5.68156756273905544654243701281, −3.24596228157815470525803297471, −2.69344253421740060260963411048, −0.69574672383740720267144236572, 2.81722780940183651087035167616, 4.43461389151825086163542395513, 5.56648847012034279232249527261, 6.59222630286824957988903006766, 7.77605578153751138394386658074, 8.968021778772664323351546338913, 9.810162774617688349816783597006, 10.73324584061007040187580580810, 12.11146368358208547865487027974, 12.92325625441774294692111734639

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