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Properties

Label 2-177-59.58-c10-0-19
Degree $2$
Conductor $177$
Sign $0.290 + 0.956i$
Analytic cond. $112.458$
Root an. cond. $10.6046$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 55.6i·2-s + 140.·3-s − 2.07e3·4-s − 1.65e3·5-s − 7.80e3i·6-s − 6.42e3·7-s + 5.84e4i·8-s + 1.96e4·9-s + 9.19e4i·10-s − 4.23e4i·11-s − 2.90e5·12-s − 1.79e5i·13-s + 3.57e5i·14-s − 2.31e5·15-s + 1.12e6·16-s − 2.44e6·17-s + ⋯
L(s)  = 1  − 1.73i·2-s + 0.577·3-s − 2.02·4-s − 0.528·5-s − 1.00i·6-s − 0.382·7-s + 1.78i·8-s + 0.333·9-s + 0.919i·10-s − 0.263i·11-s − 1.16·12-s − 0.483i·13-s + 0.665i·14-s − 0.305·15-s + 1.07·16-s − 1.72·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.290 + 0.956i$
Analytic conductor: \(112.458\)
Root analytic conductor: \(10.6046\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5),\ 0.290 + 0.956i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.009203232\)
\(L(\frac12)\) \(\approx\) \(1.009203232\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 140.T \)
59 \( 1 + (-2.07e8 - 6.84e8i)T \)
good2 \( 1 + 55.6iT - 1.02e3T^{2} \)
5 \( 1 + 1.65e3T + 9.76e6T^{2} \)
7 \( 1 + 6.42e3T + 2.82e8T^{2} \)
11 \( 1 + 4.23e4iT - 2.59e10T^{2} \)
13 \( 1 + 1.79e5iT - 1.37e11T^{2} \)
17 \( 1 + 2.44e6T + 2.01e12T^{2} \)
19 \( 1 + 2.94e6T + 6.13e12T^{2} \)
23 \( 1 - 4.47e6iT - 4.14e13T^{2} \)
29 \( 1 - 3.67e7T + 4.20e14T^{2} \)
31 \( 1 - 5.95e5iT - 8.19e14T^{2} \)
37 \( 1 + 1.07e7iT - 4.80e15T^{2} \)
41 \( 1 - 1.47e8T + 1.34e16T^{2} \)
43 \( 1 - 2.36e8iT - 2.16e16T^{2} \)
47 \( 1 + 3.78e8iT - 5.25e16T^{2} \)
53 \( 1 + 2.09e8T + 1.74e17T^{2} \)
61 \( 1 + 3.23e8iT - 7.13e17T^{2} \)
67 \( 1 + 1.67e9iT - 1.82e18T^{2} \)
71 \( 1 - 1.42e9T + 3.25e18T^{2} \)
73 \( 1 - 2.75e9iT - 4.29e18T^{2} \)
79 \( 1 - 4.89e9T + 9.46e18T^{2} \)
83 \( 1 + 4.90e8iT - 1.55e19T^{2} \)
89 \( 1 + 6.61e9iT - 3.11e19T^{2} \)
97 \( 1 + 1.12e9iT - 7.37e19T^{2} \)
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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

−10.76755148421231995235653183131, −9.823823376165555460946501582174, −8.891245505520233082844595981385, −8.090744958360385847066960283350, −6.49979422489348488723703153200, −4.65722758708271381823331006007, −3.85616055782328612788522705376, −2.85902591841170916724929882774, −1.99621680597745883791387745083, −0.65033970999140527366502888845, 0.30463590942913296715016333898, 2.33895901063477442004186369517, 4.08121030992425533135291409047, 4.67002070207121164589919173329, 6.32956953128985109109049620433, 6.81473467699680240254491667537, 7.969614668846210004858011347119, 8.664801284386953573570228297641, 9.476441809905954616662272309923, 10.85375993978904956647709802289

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