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Properties

Label 2-177-59.58-c6-0-56
Degree $2$
Conductor $177$
Sign $0.220 - 0.975i$
Analytic cond. $40.7195$
Root an. cond. $6.38118$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 12.5i·2-s − 15.5·3-s − 93.2·4-s + 48.4·5-s + 195. i·6-s − 222.·7-s + 367. i·8-s + 243·9-s − 607. i·10-s − 2.57e3i·11-s + 1.45e3·12-s − 4.16e3i·13-s + 2.78e3i·14-s − 755.·15-s − 1.36e3·16-s + 5.17e3·17-s + ⋯
L(s)  = 1  − 1.56i·2-s − 0.577·3-s − 1.45·4-s + 0.387·5-s + 0.905i·6-s − 0.647·7-s + 0.716i·8-s + 0.333·9-s − 0.607i·10-s − 1.93i·11-s + 0.841·12-s − 1.89i·13-s + 1.01i·14-s − 0.223·15-s − 0.333·16-s + 1.05·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.220 - 0.975i$
Analytic conductor: \(40.7195\)
Root analytic conductor: \(6.38118\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3),\ 0.220 - 0.975i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.8948669025\)
\(L(\frac12)\) \(\approx\) \(0.8948669025\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 15.5T \)
59 \( 1 + (4.53e4 - 2.00e5i)T \)
good2 \( 1 + 12.5iT - 64T^{2} \)
5 \( 1 - 48.4T + 1.56e4T^{2} \)
7 \( 1 + 222.T + 1.17e5T^{2} \)
11 \( 1 + 2.57e3iT - 1.77e6T^{2} \)
13 \( 1 + 4.16e3iT - 4.82e6T^{2} \)
17 \( 1 - 5.17e3T + 2.41e7T^{2} \)
19 \( 1 - 1.32e3T + 4.70e7T^{2} \)
23 \( 1 + 3.34e3iT - 1.48e8T^{2} \)
29 \( 1 - 1.60e4T + 5.94e8T^{2} \)
31 \( 1 - 2.85e4iT - 8.87e8T^{2} \)
37 \( 1 + 5.26e4iT - 2.56e9T^{2} \)
41 \( 1 + 4.94e3T + 4.75e9T^{2} \)
43 \( 1 + 2.10e4iT - 6.32e9T^{2} \)
47 \( 1 - 5.99e4iT - 1.07e10T^{2} \)
53 \( 1 + 1.05e5T + 2.21e10T^{2} \)
61 \( 1 - 1.52e5iT - 5.15e10T^{2} \)
67 \( 1 - 1.15e5iT - 9.04e10T^{2} \)
71 \( 1 - 1.81e5T + 1.28e11T^{2} \)
73 \( 1 + 6.48e5iT - 1.51e11T^{2} \)
79 \( 1 - 5.76e5T + 2.43e11T^{2} \)
83 \( 1 + 3.16e5iT - 3.26e11T^{2} \)
89 \( 1 - 2.08e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.80e5iT - 8.32e11T^{2} \)
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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

−10.65297556601300917668579518446, −10.33580266719543664155645629772, −9.210361098593293263510135743351, −7.985174136734390000519990732475, −6.15901672359175776058874976250, −5.33975189089308198027689279114, −3.52703266916864124306710555416, −2.90483416655307722891365653811, −1.07050321395491877130908591146, −0.31943159976758211090487448063, 1.86118629295275741094832447429, 4.22786182412373649170016310831, 5.15459015317005803521937695569, 6.38335834833070576322474236400, 6.89993906726678756679462946265, 7.899292564509876937799787891542, 9.567265058121624335508319560920, 9.722699842779983144301293470819, 11.57720386280481233843490892738, 12.47189559353780094737639310386

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