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Properties

Label 2-177-59.58-c4-0-3
Degree $2$
Conductor $177$
Sign $0.999 - 0.00476i$
Analytic cond. $18.2964$
Root an. cond. $4.27743$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.10i·2-s − 5.19·3-s − 0.836·4-s − 39.6·5-s + 21.3i·6-s − 85.8·7-s − 62.2i·8-s + 27·9-s + 162. i·10-s + 68.0i·11-s + 4.34·12-s − 254. i·13-s + 352. i·14-s + 205.·15-s − 268.·16-s + 229.·17-s + ⋯
L(s)  = 1  − 1.02i·2-s − 0.577·3-s − 0.0522·4-s − 1.58·5-s + 0.592i·6-s − 1.75·7-s − 0.972i·8-s + 0.333·9-s + 1.62i·10-s + 0.562i·11-s + 0.0301·12-s − 1.50i·13-s + 1.79i·14-s + 0.915·15-s − 1.04·16-s + 0.794·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00476i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 - 0.00476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.999 - 0.00476i$
Analytic conductor: \(18.2964\)
Root analytic conductor: \(4.27743\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :2),\ 0.999 - 0.00476i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.4822195006\)
\(L(\frac12)\) \(\approx\) \(0.4822195006\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.19T \)
59 \( 1 + (-3.48e3 + 16.5i)T \)
good2 \( 1 + 4.10iT - 16T^{2} \)
5 \( 1 + 39.6T + 625T^{2} \)
7 \( 1 + 85.8T + 2.40e3T^{2} \)
11 \( 1 - 68.0iT - 1.46e4T^{2} \)
13 \( 1 + 254. iT - 2.85e4T^{2} \)
17 \( 1 - 229.T + 8.35e4T^{2} \)
19 \( 1 - 425.T + 1.30e5T^{2} \)
23 \( 1 - 954. iT - 2.79e5T^{2} \)
29 \( 1 + 396.T + 7.07e5T^{2} \)
31 \( 1 - 1.42e3iT - 9.23e5T^{2} \)
37 \( 1 + 996. iT - 1.87e6T^{2} \)
41 \( 1 + 1.02e3T + 2.82e6T^{2} \)
43 \( 1 + 2.58e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.47e3iT - 4.87e6T^{2} \)
53 \( 1 + 3.15e3T + 7.89e6T^{2} \)
61 \( 1 + 2.17e3iT - 1.38e7T^{2} \)
67 \( 1 + 667. iT - 2.01e7T^{2} \)
71 \( 1 + 2.58e3T + 2.54e7T^{2} \)
73 \( 1 + 1.20e3iT - 2.83e7T^{2} \)
79 \( 1 + 1.24e3T + 3.89e7T^{2} \)
83 \( 1 - 6.11e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.20e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.69e4iT - 8.85e7T^{2} \)
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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

−12.26062805495812084280063714451, −11.13313614966421583714938639994, −10.21762772877383768030163428034, −9.477910716005135102059121285475, −7.64448298570776710552967739225, −6.97461881728539774549241692805, −5.45543417292453163872275845664, −3.55886106050629403039283053718, −3.27997652277317326294150919517, −0.843385018689464528491963121559, 0.27591253234154202438905872666, 3.16327878793103600832628848010, 4.42322425630754846966130277429, 5.99465244694173770143981187204, 6.76581869467500873455824356118, 7.54036896196411685794266206943, 8.696235850323792194787722872337, 9.930139085124916649044105613829, 11.39129977595456530433092481126, 11.81581715442499250519658827173

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