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Properties

Label 2-177-3.2-c4-0-22
Degree $2$
Conductor $177$
Sign $-0.319 + 0.947i$
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  − 7.73i·2-s + (−8.52 − 2.87i)3-s − 43.8·4-s − 0.620i·5-s + (−22.2 + 65.9i)6-s + 71.8·7-s + 215. i·8-s + (64.4 + 49.0i)9-s − 4.79·10-s − 63.2i·11-s + (373. + 126. i)12-s + 204.·13-s − 555. i·14-s + (−1.78 + 5.28i)15-s + 963.·16-s + 443. i·17-s + ⋯
L(s)  = 1  − 1.93i·2-s + (−0.947 − 0.319i)3-s − 2.73·4-s − 0.0248i·5-s + (−0.617 + 1.83i)6-s + 1.46·7-s + 3.36i·8-s + (0.795 + 0.605i)9-s − 0.0479·10-s − 0.522i·11-s + (2.59 + 0.875i)12-s + 1.21·13-s − 2.83i·14-s + (−0.00792 + 0.0235i)15-s + 3.76·16-s + 1.53i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.319 + 0.947i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.319 + 0.947i$
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: \((2,\ 177,\ (\ :2),\ -0.319 + 0.947i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.258837812\)
\(L(\frac12)\) \(\approx\) \(1.258837812\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (8.52 + 2.87i)T \)
59 \( 1 + 453. iT \)
good2 \( 1 + 7.73iT - 16T^{2} \)
5 \( 1 + 0.620iT - 625T^{2} \)
7 \( 1 - 71.8T + 2.40e3T^{2} \)
11 \( 1 + 63.2iT - 1.46e4T^{2} \)
13 \( 1 - 204.T + 2.85e4T^{2} \)
17 \( 1 - 443. iT - 8.35e4T^{2} \)
19 \( 1 + 526.T + 1.30e5T^{2} \)
23 \( 1 - 966. iT - 2.79e5T^{2} \)
29 \( 1 - 559. iT - 7.07e5T^{2} \)
31 \( 1 - 1.27e3T + 9.23e5T^{2} \)
37 \( 1 - 362.T + 1.87e6T^{2} \)
41 \( 1 + 2.02e3iT - 2.82e6T^{2} \)
43 \( 1 + 487.T + 3.41e6T^{2} \)
47 \( 1 + 34.7iT - 4.87e6T^{2} \)
53 \( 1 - 1.64e3iT - 7.89e6T^{2} \)
61 \( 1 + 2.51e3T + 1.38e7T^{2} \)
67 \( 1 + 2.37e3T + 2.01e7T^{2} \)
71 \( 1 + 884. iT - 2.54e7T^{2} \)
73 \( 1 - 7.90e3T + 2.83e7T^{2} \)
79 \( 1 + 3.00e3T + 3.89e7T^{2} \)
83 \( 1 - 4.56e3iT - 4.74e7T^{2} \)
89 \( 1 + 2.01e3iT - 6.27e7T^{2} \)
97 \( 1 - 7.87e3T + 8.85e7T^{2} \)
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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

−11.48643093108180396513886159043, −10.88736462829261726523817694496, −10.45769022720473360243418254640, −8.787679909896149179826182566468, −8.097250838738205760081351267635, −5.92778341791775347221491323061, −4.80163003844617578844253360492, −3.78849827351260225455606092676, −1.83016729284211696518939224970, −1.09493221586094133502423362111, 0.72306622585568815157350798425, 4.54753443408235220506996248444, 4.71994593251977024007484785258, 6.12409771653237397790488313067, 6.84465062409007306687602050346, 8.080625937760840951893892370729, 8.831772955424117641271686093135, 10.15204732602088117367151521119, 11.24734398923528526586947953244, 12.53012643761737985981154416995

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