Properties

Label 2-177-1.1-c3-0-0
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  − 1.67·2-s − 3·3-s − 5.20·4-s − 6.76·5-s + 5.01·6-s − 19.0·7-s + 22.0·8-s + 9·9-s + 11.3·10-s − 65.4·11-s + 15.6·12-s − 46.8·13-s + 31.8·14-s + 20.2·15-s + 4.65·16-s + 48.0·17-s − 15.0·18-s + 147.·19-s + 35.1·20-s + 57.1·21-s + 109.·22-s + 33.1·23-s − 66.2·24-s − 79.2·25-s + 78.4·26-s − 27·27-s + 99.0·28-s + ⋯
L(s)  = 1  − 0.591·2-s − 0.577·3-s − 0.650·4-s − 0.604·5-s + 0.341·6-s − 1.02·7-s + 0.976·8-s + 0.333·9-s + 0.357·10-s − 1.79·11-s + 0.375·12-s − 1.00·13-s + 0.608·14-s + 0.349·15-s + 0.0727·16-s + 0.685·17-s − 0.197·18-s + 1.77·19-s + 0.393·20-s + 0.593·21-s + 1.06·22-s + 0.300·23-s − 0.563·24-s − 0.634·25-s + 0.591·26-s − 0.192·27-s + 0.668·28-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

\(L(2)\) \(\approx\) \(0.3673603931\)
\(L(\frac12)\) \(\approx\) \(0.3673603931\)
\(L(\frac{5}{2})\) 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 \( 1 + 1.67T + 8T^{2} \)
5 \( 1 + 6.76T + 125T^{2} \)
7 \( 1 + 19.0T + 343T^{2} \)
11 \( 1 + 65.4T + 1.33e3T^{2} \)
13 \( 1 + 46.8T + 2.19e3T^{2} \)
17 \( 1 - 48.0T + 4.91e3T^{2} \)
19 \( 1 - 147.T + 6.85e3T^{2} \)
23 \( 1 - 33.1T + 1.21e4T^{2} \)
29 \( 1 + 73.6T + 2.43e4T^{2} \)
31 \( 1 - 142.T + 2.97e4T^{2} \)
37 \( 1 - 397.T + 5.06e4T^{2} \)
41 \( 1 + 100.T + 6.89e4T^{2} \)
43 \( 1 - 388.T + 7.95e4T^{2} \)
47 \( 1 + 138.T + 1.03e5T^{2} \)
53 \( 1 + 439.T + 1.48e5T^{2} \)
61 \( 1 + 602.T + 2.26e5T^{2} \)
67 \( 1 + 154.T + 3.00e5T^{2} \)
71 \( 1 - 552.T + 3.57e5T^{2} \)
73 \( 1 + 107.T + 3.89e5T^{2} \)
79 \( 1 - 989.T + 4.93e5T^{2} \)
83 \( 1 + 730.T + 5.71e5T^{2} \)
89 \( 1 + 1.37e3T + 7.04e5T^{2} \)
97 \( 1 + 268.T + 9.12e5T^{2} \)
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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

−12.31598702953902330189838276105, −11.09732126825105027760538073883, −9.920926305237601117865273576691, −9.608555270097395498841415858668, −7.87787655212124141879794306119, −7.46013904173703183755788536604, −5.68131609833118699741180171655, −4.69557533441361376736538183325, −3.08308300835040928772366291778, −0.51679212708775744164073075212, 0.51679212708775744164073075212, 3.08308300835040928772366291778, 4.69557533441361376736538183325, 5.68131609833118699741180171655, 7.46013904173703183755788536604, 7.87787655212124141879794306119, 9.608555270097395498841415858668, 9.920926305237601117865273576691, 11.09732126825105027760538073883, 12.31598702953902330189838276105

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