Properties

Degree $2$
Conductor $177$
Sign $-0.954 - 0.298i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.93·2-s + (−1.34 − 1.09i)3-s + 1.74·4-s − 3.21i·5-s + (2.59 + 2.12i)6-s + 0.254·7-s + 0.491·8-s + (0.594 + 2.94i)9-s + 6.23i·10-s − 5.68·11-s + (−2.34 − 1.91i)12-s − 2.66i·13-s − 0.491·14-s + (−3.53 + 4.31i)15-s − 4.44·16-s + 4.03i·17-s + ⋯
L(s)  = 1  − 1.36·2-s + (−0.774 − 0.633i)3-s + 0.872·4-s − 1.43i·5-s + (1.05 + 0.866i)6-s + 0.0960·7-s + 0.173·8-s + (0.198 + 0.980i)9-s + 1.97i·10-s − 1.71·11-s + (−0.675 − 0.552i)12-s − 0.738i·13-s − 0.131·14-s + (−0.911 + 1.11i)15-s − 1.11·16-s + 0.979i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.954 - 0.298i$
Motivic weight: \(1\)
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ -0.954 - 0.298i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0250473 + 0.164212i\)
\(L(\frac12)\) \(\approx\) \(0.0250473 + 0.164212i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.34 + 1.09i)T \)
59 \( 1 + (7.12 - 2.86i)T \)
good2 \( 1 + 1.93T + 2T^{2} \)
5 \( 1 + 3.21iT - 5T^{2} \)
7 \( 1 - 0.254T + 7T^{2} \)
11 \( 1 + 5.68T + 11T^{2} \)
13 \( 1 + 2.66iT - 13T^{2} \)
17 \( 1 - 4.03iT - 17T^{2} \)
19 \( 1 + 2.44T + 19T^{2} \)
23 \( 1 + 3.25T + 23T^{2} \)
29 \( 1 - 3.56iT - 29T^{2} \)
31 \( 1 - 7.81iT - 31T^{2} \)
37 \( 1 + 5.15iT - 37T^{2} \)
41 \( 1 + 7.25iT - 41T^{2} \)
43 \( 1 + 9.79iT - 43T^{2} \)
47 \( 1 - 5.06T + 47T^{2} \)
53 \( 1 - 1.16iT - 53T^{2} \)
61 \( 1 - 0.676iT - 61T^{2} \)
67 \( 1 - 2.66iT - 67T^{2} \)
71 \( 1 + 16.2iT - 71T^{2} \)
73 \( 1 + 13.1iT - 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 + 9.91T + 83T^{2} \)
89 \( 1 - 7.95T + 89T^{2} \)
97 \( 1 + 4.47iT - 97T^{2} \)
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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

−12.29047589582257049487408185201, −10.65547283448789191658143575265, −10.42175789752673388561630437532, −8.831689904844680518105288663144, −8.179177154598961222650854187464, −7.39223887242312872742745910537, −5.71879654441117092135157641968, −4.79549447942568062278911336079, −1.80939368130398762733752978900, −0.25028104011450143750063096256, 2.62906071688744704892780818425, 4.55846289460673899735296373355, 6.14658450996326248975696372143, 7.19328153956398776061272954335, 8.137538188671499054736771187762, 9.713439522133988979126753756812, 10.06632599297428783908253146625, 11.10190229511788420708012132897, 11.45195241698577571222353149920, 13.19202070856578911557366925179

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