Properties

Label 2-177-1.1-c3-0-21
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.77·2-s + 3·3-s − 4.84·4-s + 6.23·5-s − 5.32·6-s − 18.0·7-s + 22.8·8-s + 9·9-s − 11.0·10-s + 13.3·11-s − 14.5·12-s − 66.3·13-s + 32.0·14-s + 18.6·15-s − 1.68·16-s − 97.6·17-s − 15.9·18-s + 109.·19-s − 30.2·20-s − 54.2·21-s − 23.6·22-s − 147.·23-s + 68.4·24-s − 86.1·25-s + 117.·26-s + 27·27-s + 87.6·28-s + ⋯
L(s)  = 1  − 0.627·2-s + 0.577·3-s − 0.606·4-s + 0.557·5-s − 0.362·6-s − 0.976·7-s + 1.00·8-s + 0.333·9-s − 0.349·10-s + 0.365·11-s − 0.349·12-s − 1.41·13-s + 0.612·14-s + 0.321·15-s − 0.0263·16-s − 1.39·17-s − 0.209·18-s + 1.32·19-s − 0.337·20-s − 0.563·21-s − 0.229·22-s − 1.33·23-s + 0.581·24-s − 0.689·25-s + 0.888·26-s + 0.192·27-s + 0.591·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 1.77T + 8T^{2} \)
5 \( 1 - 6.23T + 125T^{2} \)
7 \( 1 + 18.0T + 343T^{2} \)
11 \( 1 - 13.3T + 1.33e3T^{2} \)
13 \( 1 + 66.3T + 2.19e3T^{2} \)
17 \( 1 + 97.6T + 4.91e3T^{2} \)
19 \( 1 - 109.T + 6.85e3T^{2} \)
23 \( 1 + 147.T + 1.21e4T^{2} \)
29 \( 1 + 173.T + 2.43e4T^{2} \)
31 \( 1 - 148.T + 2.97e4T^{2} \)
37 \( 1 + 446.T + 5.06e4T^{2} \)
41 \( 1 - 182.T + 6.89e4T^{2} \)
43 \( 1 + 223.T + 7.95e4T^{2} \)
47 \( 1 + 529.T + 1.03e5T^{2} \)
53 \( 1 - 398.T + 1.48e5T^{2} \)
61 \( 1 - 788.T + 2.26e5T^{2} \)
67 \( 1 - 288.T + 3.00e5T^{2} \)
71 \( 1 + 139.T + 3.57e5T^{2} \)
73 \( 1 - 549.T + 3.89e5T^{2} \)
79 \( 1 + 190.T + 4.93e5T^{2} \)
83 \( 1 - 410.T + 5.71e5T^{2} \)
89 \( 1 - 1.29e3T + 7.04e5T^{2} \)
97 \( 1 - 1.48e3T + 9.12e5T^{2} \)
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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

−11.78418271443563426370521481313, −10.07829152469800530248039957614, −9.734208686093020312042537359052, −8.942362279301094827046087176026, −7.75265052060360502413297237569, −6.68340614886887603538688128090, −5.12993932603101210106068055557, −3.72752084068713117923020223843, −2.06797750947095367109343158962, 0, 2.06797750947095367109343158962, 3.72752084068713117923020223843, 5.12993932603101210106068055557, 6.68340614886887603538688128090, 7.75265052060360502413297237569, 8.942362279301094827046087176026, 9.734208686093020312042537359052, 10.07829152469800530248039957614, 11.78418271443563426370521481313

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