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Properties

Label 2-177-3.2-c4-0-21
Degree $2$
Conductor $177$
Sign $-0.196 + 0.980i$
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  − 6.47i·2-s + (−8.82 − 1.76i)3-s − 25.9·4-s + 26.7i·5-s + (−11.4 + 57.1i)6-s − 48.5·7-s + 64.2i·8-s + (74.7 + 31.1i)9-s + 173.·10-s + 66.3i·11-s + (228. + 45.7i)12-s − 17.1·13-s + 314. i·14-s + (47.2 − 236. i)15-s + 1.17·16-s − 329. i·17-s + ⋯
L(s)  = 1  − 1.61i·2-s + (−0.980 − 0.196i)3-s − 1.62·4-s + 1.07i·5-s + (−0.317 + 1.58i)6-s − 0.990·7-s + 1.00i·8-s + (0.923 + 0.384i)9-s + 1.73·10-s + 0.548i·11-s + (1.58 + 0.317i)12-s − 0.101·13-s + 1.60i·14-s + (0.209 − 1.04i)15-s + 0.00459·16-s − 1.13i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.196 + 0.980i$
Analytic conductor: \(18.2964\)
Root analytic conductor: \(4.27743\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :2),\ -0.196 + 0.980i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.9084215185\)
\(L(\frac12)\) \(\approx\) \(0.9084215185\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (8.82 + 1.76i)T \)
59 \( 1 + 453. iT \)
good2 \( 1 + 6.47iT - 16T^{2} \)
5 \( 1 - 26.7iT - 625T^{2} \)
7 \( 1 + 48.5T + 2.40e3T^{2} \)
11 \( 1 - 66.3iT - 1.46e4T^{2} \)
13 \( 1 + 17.1T + 2.85e4T^{2} \)
17 \( 1 + 329. iT - 8.35e4T^{2} \)
19 \( 1 - 308.T + 1.30e5T^{2} \)
23 \( 1 + 68.4iT - 2.79e5T^{2} \)
29 \( 1 - 178. iT - 7.07e5T^{2} \)
31 \( 1 - 80.5T + 9.23e5T^{2} \)
37 \( 1 - 1.12e3T + 1.87e6T^{2} \)
41 \( 1 - 426. iT - 2.82e6T^{2} \)
43 \( 1 - 1.03e3T + 3.41e6T^{2} \)
47 \( 1 + 574. iT - 4.87e6T^{2} \)
53 \( 1 - 409. iT - 7.89e6T^{2} \)
61 \( 1 - 4.08e3T + 1.38e7T^{2} \)
67 \( 1 - 8.02e3T + 2.01e7T^{2} \)
71 \( 1 + 9.53e3iT - 2.54e7T^{2} \)
73 \( 1 - 6.39e3T + 2.83e7T^{2} \)
79 \( 1 + 2.58e3T + 3.89e7T^{2} \)
83 \( 1 + 3.76e3iT - 4.74e7T^{2} \)
89 \( 1 - 5.96e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.53e4T + 8.85e7T^{2} \)
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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

−11.61278406963802567118839001427, −10.91298914874719538402001089184, −9.999504558802736349076303247515, −9.485363238991237423099230699260, −7.32836048417003100100302892426, −6.46321054571701494686460951792, −4.91699113785504248823036793753, −3.48251323745040365479428637593, −2.39948547095320639435009014141, −0.67700801823486086838160736425, 0.70568326236162847390487810592, 3.99698721512336360534658165639, 5.20064666413139058776436889017, 5.93408336663716022840434949619, 6.79828761956396180763022283002, 8.029488339135978315722947429329, 9.081560251097082802437595877089, 9.949856677214935212276505106216, 11.39142503193672473479096889443, 12.66691918974839478829632548296

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