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Properties

Label 2-177-59.58-c2-0-16
Degree $2$
Conductor $177$
Sign $-0.0793 + 0.996i$
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  − 1.61i·2-s + 1.73·3-s + 1.39·4-s − 6.36·5-s − 2.79i·6-s + 6.66·7-s − 8.70i·8-s + 2.99·9-s + 10.2i·10-s − 16.0i·11-s + 2.41·12-s − 7.84i·13-s − 10.7i·14-s − 11.0·15-s − 8.46·16-s + 18.9·17-s + ⋯
L(s)  = 1  − 0.806i·2-s + 0.577·3-s + 0.349·4-s − 1.27·5-s − 0.465i·6-s + 0.952·7-s − 1.08i·8-s + 0.333·9-s + 1.02i·10-s − 1.45i·11-s + 0.201·12-s − 0.603i·13-s − 0.768i·14-s − 0.735·15-s − 0.529·16-s + 1.11·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0793 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0793 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.0793 + 0.996i$
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: \((2,\ 177,\ (\ :1),\ -0.0793 + 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.24100 - 1.34368i\)
\(L(\frac12)\) \(\approx\) \(1.24100 - 1.34368i\)
\(L(2)\) 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 \( 1 + (4.68 - 58.8i)T \)
good2 \( 1 + 1.61iT - 4T^{2} \)
5 \( 1 + 6.36T + 25T^{2} \)
7 \( 1 - 6.66T + 49T^{2} \)
11 \( 1 + 16.0iT - 121T^{2} \)
13 \( 1 + 7.84iT - 169T^{2} \)
17 \( 1 - 18.9T + 289T^{2} \)
19 \( 1 - 7.10T + 361T^{2} \)
23 \( 1 - 33.6iT - 529T^{2} \)
29 \( 1 + 46.2T + 841T^{2} \)
31 \( 1 - 29.5iT - 961T^{2} \)
37 \( 1 - 1.91iT - 1.36e3T^{2} \)
41 \( 1 - 46.4T + 1.68e3T^{2} \)
43 \( 1 + 21.6iT - 1.84e3T^{2} \)
47 \( 1 - 75.3iT - 2.20e3T^{2} \)
53 \( 1 - 19.5T + 2.80e3T^{2} \)
61 \( 1 - 41.0iT - 3.72e3T^{2} \)
67 \( 1 - 90.1iT - 4.48e3T^{2} \)
71 \( 1 - 57.1T + 5.04e3T^{2} \)
73 \( 1 + 69.6iT - 5.32e3T^{2} \)
79 \( 1 - 118.T + 6.24e3T^{2} \)
83 \( 1 + 86.3iT - 6.88e3T^{2} \)
89 \( 1 + 38.4iT - 7.92e3T^{2} \)
97 \( 1 - 24.0iT - 9.40e3T^{2} \)
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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

−11.87903569803177837020619501095, −11.34025262791339684393601556207, −10.57172085850394418586407271947, −9.183462725900334678028539762824, −7.910359817548377773085869857808, −7.51225537006502385060683388662, −5.60733922338435676155099279386, −3.82907378393864029670589365861, −3.11112729745308283522005307351, −1.17879301064979186212655335437, 2.10630613753167731574607459577, 3.99067210117823045466533304093, 5.09421382067701072869192188616, 6.82277903878506379160176334947, 7.73416124236891849772341166223, 8.066235177868574353558200576861, 9.483978429051284365218882714038, 10.92575789247868623445569635276, 11.77533017056086992673845810258, 12.54767214659243380284480763658

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