Properties

Label 2-177-59.58-c2-0-14
Degree $2$
Conductor $177$
Sign $0.738 + 0.674i$
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  − 1.08i·2-s + 1.73·3-s + 2.82·4-s + 3.33·5-s − 1.88i·6-s − 1.22·7-s − 7.40i·8-s + 2.99·9-s − 3.62i·10-s + 1.21i·11-s + 4.88·12-s + 12.6i·13-s + 1.33i·14-s + 5.77·15-s + 3.23·16-s − 0.815·17-s + ⋯
L(s)  = 1  − 0.543i·2-s + 0.577·3-s + 0.705·4-s + 0.667·5-s − 0.313i·6-s − 0.175·7-s − 0.926i·8-s + 0.333·9-s − 0.362i·10-s + 0.110i·11-s + 0.407·12-s + 0.973i·13-s + 0.0951i·14-s + 0.385·15-s + 0.202·16-s − 0.0479·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 + 0.674i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.738 + 0.674i$
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: \((2,\ 177,\ (\ :1),\ 0.738 + 0.674i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.11221 - 0.819372i\)
\(L(\frac12)\) \(\approx\) \(2.11221 - 0.819372i\)
\(L(2)\) 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 \( 1 + (-43.5 - 39.7i)T \)
good2 \( 1 + 1.08iT - 4T^{2} \)
5 \( 1 - 3.33T + 25T^{2} \)
7 \( 1 + 1.22T + 49T^{2} \)
11 \( 1 - 1.21iT - 121T^{2} \)
13 \( 1 - 12.6iT - 169T^{2} \)
17 \( 1 + 0.815T + 289T^{2} \)
19 \( 1 + 7.74T + 361T^{2} \)
23 \( 1 + 33.8iT - 529T^{2} \)
29 \( 1 - 7.53T + 841T^{2} \)
31 \( 1 - 7.71iT - 961T^{2} \)
37 \( 1 - 16.7iT - 1.36e3T^{2} \)
41 \( 1 + 19.0T + 1.68e3T^{2} \)
43 \( 1 - 48.9iT - 1.84e3T^{2} \)
47 \( 1 + 4.79iT - 2.20e3T^{2} \)
53 \( 1 + 48.7T + 2.80e3T^{2} \)
61 \( 1 - 87.6iT - 3.72e3T^{2} \)
67 \( 1 + 42.6iT - 4.48e3T^{2} \)
71 \( 1 + 20.2T + 5.04e3T^{2} \)
73 \( 1 - 45.6iT - 5.32e3T^{2} \)
79 \( 1 + 62.5T + 6.24e3T^{2} \)
83 \( 1 - 22.8iT - 6.88e3T^{2} \)
89 \( 1 - 55.8iT - 7.92e3T^{2} \)
97 \( 1 + 123. iT - 9.40e3T^{2} \)
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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

−12.35385436758185334605935673133, −11.34432771872489048141344801855, −10.28205960776602600706559318279, −9.564339845759984507623845535835, −8.400454638749779114287110341368, −6.99840193649230692261079108949, −6.18678475173206793428193445977, −4.34672924578781703479158609772, −2.84453489609604153094783936196, −1.73674660453207865961493210556, 1.96778098985254429067853212716, 3.33486054044267512586986201175, 5.32266162283151503615821112364, 6.27094145317189930748600514934, 7.43275502295935998370624596524, 8.295952892299862609063649648656, 9.537985485507902845207156286533, 10.47434177816103645800723889937, 11.55616947273567691258980660410, 12.77597855822849655511433618105

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