Properties

Label 2-177-59.58-c4-0-19
Degree $2$
Conductor $177$
Sign $0.960 + 0.277i$
Analytic cond. $18.2964$
Root an. cond. $4.27743$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.15i·2-s − 5.19·3-s + 11.3·4-s + 30.9·5-s + 11.2i·6-s + 73.3·7-s − 58.9i·8-s + 27·9-s − 66.6i·10-s + 203. i·11-s − 58.9·12-s + 115. i·13-s − 158. i·14-s − 160.·15-s + 54.5·16-s − 149.·17-s + ⋯
L(s)  = 1  − 0.538i·2-s − 0.577·3-s + 0.709·4-s + 1.23·5-s + 0.311i·6-s + 1.49·7-s − 0.921i·8-s + 0.333·9-s − 0.666i·10-s + 1.68i·11-s − 0.409·12-s + 0.681i·13-s − 0.806i·14-s − 0.714·15-s + 0.213·16-s − 0.517·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.960 + 0.277i$
Analytic conductor: \(18.2964\)
Root analytic conductor: \(4.27743\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :2),\ 0.960 + 0.277i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.725010205\)
\(L(\frac12)\) \(\approx\) \(2.725010205\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.19T \)
59 \( 1 + (-3.34e3 - 964. i)T \)
good2 \( 1 + 2.15iT - 16T^{2} \)
5 \( 1 - 30.9T + 625T^{2} \)
7 \( 1 - 73.3T + 2.40e3T^{2} \)
11 \( 1 - 203. iT - 1.46e4T^{2} \)
13 \( 1 - 115. iT - 2.85e4T^{2} \)
17 \( 1 + 149.T + 8.35e4T^{2} \)
19 \( 1 + 97.5T + 1.30e5T^{2} \)
23 \( 1 - 785. iT - 2.79e5T^{2} \)
29 \( 1 - 279.T + 7.07e5T^{2} \)
31 \( 1 + 520. iT - 9.23e5T^{2} \)
37 \( 1 + 737. iT - 1.87e6T^{2} \)
41 \( 1 - 540.T + 2.82e6T^{2} \)
43 \( 1 + 1.17e3iT - 3.41e6T^{2} \)
47 \( 1 + 1.50e3iT - 4.87e6T^{2} \)
53 \( 1 + 3.63e3T + 7.89e6T^{2} \)
61 \( 1 + 5.34e3iT - 1.38e7T^{2} \)
67 \( 1 + 1.69e3iT - 2.01e7T^{2} \)
71 \( 1 + 9.06e3T + 2.54e7T^{2} \)
73 \( 1 - 876. iT - 2.83e7T^{2} \)
79 \( 1 - 7.89e3T + 3.89e7T^{2} \)
83 \( 1 + 3.53e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.14e4iT - 6.27e7T^{2} \)
97 \( 1 + 1.51e4iT - 8.85e7T^{2} \)
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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

−11.77861283691194148030354540763, −11.09175147366416862512834900494, −10.12954707957475856401990599609, −9.353880462393328705419348937879, −7.61108269463670313010450222958, −6.68697704716295923817858442816, −5.46308779297468250210203529201, −4.36376629531641198804428574897, −2.03202839508454615828539318389, −1.66494174373677407925847717187, 1.23023551475726506651298933362, 2.58121780474787656346136911357, 4.89564772815068459986381528897, 5.81702695992036129728186742574, 6.47518533040311232712689086671, 7.953996759679657297707328257392, 8.723155067321634859949499126802, 10.47693301141330455760702510958, 10.93814085320820012144750916385, 11.82344409771297229678444571040

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