Properties

Label 2-177-59.58-c2-0-9
Degree $2$
Conductor $177$
Sign $-0.314 - 0.949i$
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  + 2.96i·2-s + 1.73·3-s − 4.77·4-s + 3.25·5-s + 5.13i·6-s + 11.8·7-s − 2.31i·8-s + 2.99·9-s + 9.63i·10-s − 16.1i·11-s − 8.27·12-s + 14.1i·13-s + 35.0i·14-s + 5.63·15-s − 12.2·16-s − 32.0·17-s + ⋯
L(s)  = 1  + 1.48i·2-s + 0.577·3-s − 1.19·4-s + 0.650·5-s + 0.855i·6-s + 1.68·7-s − 0.288i·8-s + 0.333·9-s + 0.963i·10-s − 1.46i·11-s − 0.689·12-s + 1.08i·13-s + 2.50i·14-s + 0.375·15-s − 0.767·16-s − 1.88·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.314 - 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.314 - 0.949i$
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: \((2,\ 177,\ (\ :1),\ -0.314 - 0.949i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.24782 + 1.72883i\)
\(L(\frac12)\) \(\approx\) \(1.24782 + 1.72883i\)
\(L(2)\) 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 \( 1 + (18.5 + 55.9i)T \)
good2 \( 1 - 2.96iT - 4T^{2} \)
5 \( 1 - 3.25T + 25T^{2} \)
7 \( 1 - 11.8T + 49T^{2} \)
11 \( 1 + 16.1iT - 121T^{2} \)
13 \( 1 - 14.1iT - 169T^{2} \)
17 \( 1 + 32.0T + 289T^{2} \)
19 \( 1 - 14.7T + 361T^{2} \)
23 \( 1 + 14.5iT - 529T^{2} \)
29 \( 1 + 12.7T + 841T^{2} \)
31 \( 1 - 27.0iT - 961T^{2} \)
37 \( 1 + 26.2iT - 1.36e3T^{2} \)
41 \( 1 + 37.3T + 1.68e3T^{2} \)
43 \( 1 + 51.1iT - 1.84e3T^{2} \)
47 \( 1 - 79.6iT - 2.20e3T^{2} \)
53 \( 1 - 86.2T + 2.80e3T^{2} \)
61 \( 1 - 92.8iT - 3.72e3T^{2} \)
67 \( 1 + 88.8iT - 4.48e3T^{2} \)
71 \( 1 + 13.4T + 5.04e3T^{2} \)
73 \( 1 + 42.9iT - 5.32e3T^{2} \)
79 \( 1 + 1.74T + 6.24e3T^{2} \)
83 \( 1 - 12.9iT - 6.88e3T^{2} \)
89 \( 1 + 69.4iT - 7.92e3T^{2} \)
97 \( 1 + 101. iT - 9.40e3T^{2} \)
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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

−13.59069534139072182158087124618, −11.61740398162432643800039225793, −10.83976124604174460294552206704, −8.977183536491310942568987048604, −8.682253440556332469429740368654, −7.60638010212524730588163087709, −6.53522513273449767809945893063, −5.41589762938649687067078119329, −4.35783504505714503135473785508, −2.03545393300132907733747254476, 1.63729299256276618711374473467, 2.39317856441518107229181598514, 4.14727250183133274578128715106, 5.14510069064212934076530054105, 7.19831429349444459553555598959, 8.359275035701798384735062434540, 9.496092625593652786297067808090, 10.22359289885862887735169566469, 11.23028959291842250567538389270, 11.95651558333461237376553936237

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