Properties

Label 2-177-1.1-c3-0-3
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.780·2-s − 3·3-s − 7.39·4-s − 21.9·5-s − 2.34·6-s + 32.1·7-s − 12.0·8-s + 9·9-s − 17.0·10-s − 35.3·11-s + 22.1·12-s + 34.5·13-s + 25.0·14-s + 65.7·15-s + 49.7·16-s − 27.9·17-s + 7.02·18-s + 74.3·19-s + 162.·20-s − 96.3·21-s − 27.5·22-s + 128.·23-s + 36.0·24-s + 355.·25-s + 26.9·26-s − 27·27-s − 237.·28-s + ⋯
L(s)  = 1  + 0.275·2-s − 0.577·3-s − 0.923·4-s − 1.96·5-s − 0.159·6-s + 1.73·7-s − 0.530·8-s + 0.333·9-s − 0.540·10-s − 0.967·11-s + 0.533·12-s + 0.736·13-s + 0.478·14-s + 1.13·15-s + 0.777·16-s − 0.399·17-s + 0.0919·18-s + 0.897·19-s + 1.81·20-s − 1.00·21-s − 0.266·22-s + 1.16·23-s + 0.306·24-s + 2.84·25-s + 0.203·26-s − 0.192·27-s − 1.60·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9191737869\)
\(L(\frac12)\) \(\approx\) \(0.9191737869\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
59 \( 1 + 59T \)
good2 \( 1 - 0.780T + 8T^{2} \)
5 \( 1 + 21.9T + 125T^{2} \)
7 \( 1 - 32.1T + 343T^{2} \)
11 \( 1 + 35.3T + 1.33e3T^{2} \)
13 \( 1 - 34.5T + 2.19e3T^{2} \)
17 \( 1 + 27.9T + 4.91e3T^{2} \)
19 \( 1 - 74.3T + 6.85e3T^{2} \)
23 \( 1 - 128.T + 1.21e4T^{2} \)
29 \( 1 - 42.7T + 2.43e4T^{2} \)
31 \( 1 + 211.T + 2.97e4T^{2} \)
37 \( 1 + 165.T + 5.06e4T^{2} \)
41 \( 1 - 377.T + 6.89e4T^{2} \)
43 \( 1 + 393.T + 7.95e4T^{2} \)
47 \( 1 - 261.T + 1.03e5T^{2} \)
53 \( 1 + 113.T + 1.48e5T^{2} \)
61 \( 1 - 337.T + 2.26e5T^{2} \)
67 \( 1 - 183.T + 3.00e5T^{2} \)
71 \( 1 + 168.T + 3.57e5T^{2} \)
73 \( 1 - 805.T + 3.89e5T^{2} \)
79 \( 1 - 797.T + 4.93e5T^{2} \)
83 \( 1 - 251.T + 5.71e5T^{2} \)
89 \( 1 - 653.T + 7.04e5T^{2} \)
97 \( 1 + 1.57e3T + 9.12e5T^{2} \)
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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

−12.11824608556285494997266444973, −11.25602678153488652485473977365, −10.77831674649618362174415066783, −8.848434346641670044491190720846, −8.080171826592934420480470558090, −7.33806983536757041267130994978, −5.24818082800834043213733079606, −4.67173796312431542786644436872, −3.57837690729454845491835046412, −0.76509155334516902874932818328, 0.76509155334516902874932818328, 3.57837690729454845491835046412, 4.67173796312431542786644436872, 5.24818082800834043213733079606, 7.33806983536757041267130994978, 8.080171826592934420480470558090, 8.848434346641670044491190720846, 10.77831674649618362174415066783, 11.25602678153488652485473977365, 12.11824608556285494997266444973

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