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Properties

Label 2-177-59.58-c2-0-0
Degree $2$
Conductor $177$
Sign $0.703 - 0.710i$
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  − 3.65i·2-s − 1.73·3-s − 9.37·4-s + 0.395·5-s + 6.33i·6-s − 6.20·7-s + 19.6i·8-s + 2.99·9-s − 1.44i·10-s + 8.54i·11-s + 16.2·12-s + 0.0887i·13-s + 22.7i·14-s − 0.685·15-s + 34.4·16-s − 8.10·17-s + ⋯
L(s)  = 1  − 1.82i·2-s − 0.577·3-s − 2.34·4-s + 0.0791·5-s + 1.05i·6-s − 0.886·7-s + 2.45i·8-s + 0.333·9-s − 0.144i·10-s + 0.777i·11-s + 1.35·12-s + 0.00682i·13-s + 1.62i·14-s − 0.0457·15-s + 2.15·16-s − 0.477·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.703 - 0.710i$
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: \((2,\ 177,\ (\ :1),\ 0.703 - 0.710i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0865935 + 0.0360961i\)
\(L(\frac12)\) \(\approx\) \(0.0865935 + 0.0360961i\)
\(L(2)\) 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 \( 1 + (41.5 - 41.9i)T \)
good2 \( 1 + 3.65iT - 4T^{2} \)
5 \( 1 - 0.395T + 25T^{2} \)
7 \( 1 + 6.20T + 49T^{2} \)
11 \( 1 - 8.54iT - 121T^{2} \)
13 \( 1 - 0.0887iT - 169T^{2} \)
17 \( 1 + 8.10T + 289T^{2} \)
19 \( 1 - 10.5T + 361T^{2} \)
23 \( 1 - 14.0iT - 529T^{2} \)
29 \( 1 + 56.9T + 841T^{2} \)
31 \( 1 + 0.471iT - 961T^{2} \)
37 \( 1 + 43.8iT - 1.36e3T^{2} \)
41 \( 1 - 13.8T + 1.68e3T^{2} \)
43 \( 1 - 53.7iT - 1.84e3T^{2} \)
47 \( 1 - 34.1iT - 2.20e3T^{2} \)
53 \( 1 + 79.3T + 2.80e3T^{2} \)
61 \( 1 - 45.7iT - 3.72e3T^{2} \)
67 \( 1 + 104. iT - 4.48e3T^{2} \)
71 \( 1 - 74.0T + 5.04e3T^{2} \)
73 \( 1 + 109. iT - 5.32e3T^{2} \)
79 \( 1 + 74.2T + 6.24e3T^{2} \)
83 \( 1 + 88.8iT - 6.88e3T^{2} \)
89 \( 1 - 142. iT - 7.92e3T^{2} \)
97 \( 1 + 81.4iT - 9.40e3T^{2} \)
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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

−12.52366414879323912964342544502, −11.50543564913521898394798110157, −10.82635907351547057765183786569, −9.601089192256668780974450567738, −9.417290906848166558782465907204, −7.53956673077537789062761308022, −5.88264402555005786081604547142, −4.52152881421215239921492865864, −3.36791588932394575698787904530, −1.83025235962090972816089448903, 0.05994634259757846635793119946, 3.79080801909771837495793105514, 5.24168998509525896501959773006, 6.09431734055522395220310029550, 6.87272663380062447533465792635, 7.958155566201414429216916157634, 9.089298081352835554648623990570, 9.935947600852385540558761986645, 11.36931391304602566910356336762, 12.80717031205220259566471534654

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