Properties

Label 2-177-59.58-c4-0-29
Degree $2$
Conductor $177$
Sign $0.471 + 0.882i$
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  + 0.389i·2-s + 5.19·3-s + 15.8·4-s − 17.9·5-s + 2.02i·6-s − 47.2·7-s + 12.3i·8-s + 27·9-s − 7.00i·10-s − 197. i·11-s + 82.3·12-s − 176. i·13-s − 18.3i·14-s − 93.5·15-s + 248.·16-s + 486.·17-s + ⋯
L(s)  = 1  + 0.0972i·2-s + 0.577·3-s + 0.990·4-s − 0.719·5-s + 0.0561i·6-s − 0.964·7-s + 0.193i·8-s + 0.333·9-s − 0.0700i·10-s − 1.63i·11-s + 0.571·12-s − 1.04i·13-s − 0.0938i·14-s − 0.415·15-s + 0.971·16-s + 1.68·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.882i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.471 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.471 + 0.882i$
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: \((2,\ 177,\ (\ :2),\ 0.471 + 0.882i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.167934737\)
\(L(\frac12)\) \(\approx\) \(2.167934737\)
\(L(3)\) 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 \( 1 + (1.64e3 + 3.07e3i)T \)
good2 \( 1 - 0.389iT - 16T^{2} \)
5 \( 1 + 17.9T + 625T^{2} \)
7 \( 1 + 47.2T + 2.40e3T^{2} \)
11 \( 1 + 197. iT - 1.46e4T^{2} \)
13 \( 1 + 176. iT - 2.85e4T^{2} \)
17 \( 1 - 486.T + 8.35e4T^{2} \)
19 \( 1 - 56.2T + 1.30e5T^{2} \)
23 \( 1 + 848. iT - 2.79e5T^{2} \)
29 \( 1 - 275.T + 7.07e5T^{2} \)
31 \( 1 - 843. iT - 9.23e5T^{2} \)
37 \( 1 - 1.85e3iT - 1.87e6T^{2} \)
41 \( 1 - 1.47e3T + 2.82e6T^{2} \)
43 \( 1 + 2.75e3iT - 3.41e6T^{2} \)
47 \( 1 + 1.43e3iT - 4.87e6T^{2} \)
53 \( 1 + 1.11e3T + 7.89e6T^{2} \)
61 \( 1 + 115. iT - 1.38e7T^{2} \)
67 \( 1 + 49.5iT - 2.01e7T^{2} \)
71 \( 1 + 2.08e3T + 2.54e7T^{2} \)
73 \( 1 + 209. iT - 2.83e7T^{2} \)
79 \( 1 + 6.54e3T + 3.89e7T^{2} \)
83 \( 1 - 6.80e3iT - 4.74e7T^{2} \)
89 \( 1 - 9.11e3iT - 6.27e7T^{2} \)
97 \( 1 + 2.94e3iT - 8.85e7T^{2} \)
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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

−11.96227368449317147282186119122, −10.76504634026494972175671691077, −10.02374598477805968947790690887, −8.455985288588421511408907170737, −7.84632096173540158048070110026, −6.62835585907513802824423555247, −5.61981176122431066763337043098, −3.45589394141272657385435183092, −2.96051640509040109445353618480, −0.75327542977425079848788361952, 1.65626613254303885809169546706, 3.06277466036015598654451706631, 4.14274164139846845733046831559, 5.99934722557917350171577718208, 7.36689057572461014525262094988, 7.59509624981363470874641079847, 9.494559140510249733661831327206, 9.924207143427792161875697349826, 11.39897932849664111665764985714, 12.17263001316793562400229202053

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