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Properties

Label 2-177-177.176-c5-0-52
Degree $2$
Conductor $177$
Sign $0.879 + 0.476i$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.9·2-s + (4.58 − 14.8i)3-s + 87.6·4-s + 82.7i·5-s + (−50.1 + 162. i)6-s + 211.·7-s − 608.·8-s + (−200. − 136. i)9-s − 905. i·10-s + 575.·11-s + (402. − 1.30e3i)12-s − 586. i·13-s − 2.31e3·14-s + (1.23e3 + 379. i)15-s + 3.85e3·16-s − 230. i·17-s + ⋯
L(s)  = 1  − 1.93·2-s + (0.294 − 0.955i)3-s + 2.73·4-s + 1.48i·5-s + (−0.569 + 1.84i)6-s + 1.63·7-s − 3.36·8-s + (−0.826 − 0.562i)9-s − 2.86i·10-s + 1.43·11-s + (0.806 − 2.61i)12-s − 0.962i·13-s − 3.15·14-s + (1.41 + 0.435i)15-s + 3.76·16-s − 0.193i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.879 + 0.476i$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ 0.879 + 0.476i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.189193182\)
\(L(\frac12)\) \(\approx\) \(1.189193182\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.58 + 14.8i)T \)
59 \( 1 + (-5.25e3 + 2.62e4i)T \)
good2 \( 1 + 10.9T + 32T^{2} \)
5 \( 1 - 82.7iT - 3.12e3T^{2} \)
7 \( 1 - 211.T + 1.68e4T^{2} \)
11 \( 1 - 575.T + 1.61e5T^{2} \)
13 \( 1 + 586. iT - 3.71e5T^{2} \)
17 \( 1 + 230. iT - 1.41e6T^{2} \)
19 \( 1 - 39.4T + 2.47e6T^{2} \)
23 \( 1 - 1.71e3T + 6.43e6T^{2} \)
29 \( 1 + 7.86e3iT - 2.05e7T^{2} \)
31 \( 1 - 1.80e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.32e4iT - 6.93e7T^{2} \)
41 \( 1 + 282. iT - 1.15e8T^{2} \)
43 \( 1 - 7.61e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.25e4T + 2.29e8T^{2} \)
53 \( 1 - 9.28e3iT - 4.18e8T^{2} \)
61 \( 1 + 1.73e4iT - 8.44e8T^{2} \)
67 \( 1 + 9.81e3iT - 1.35e9T^{2} \)
71 \( 1 - 1.24e4iT - 1.80e9T^{2} \)
73 \( 1 + 5.95e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.41e3T + 3.07e9T^{2} \)
83 \( 1 - 3.94e4T + 3.93e9T^{2} \)
89 \( 1 + 5.72e4T + 5.58e9T^{2} \)
97 \( 1 + 4.08e4iT - 8.58e9T^{2} \)
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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

−11.40663059571400166264444691391, −10.77636644802955572206278212465, −9.590492988146933130676714929035, −8.412401634676924645010084861556, −7.77552541391798615952227365179, −6.96305726941754058672642420031, −6.12999862829932714633707277319, −3.03481636950142487071863963351, −1.93319097021931585242676319788, −0.903171262580953931575194077920, 1.04741535861979121994537412383, 1.89433856750653452896457531410, 4.15981912695721461691195356013, 5.45741907215417828434197356822, 7.20851324661821974832165423911, 8.446022045534026840603956778656, 8.879787025286823858240971051066, 9.375864028299529338858229411403, 10.74060457489193004678660114976, 11.45444970226545393943738978712

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