Properties

Label 2-177-1.1-c3-0-7
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  + 2.17·2-s − 3·3-s − 3.28·4-s + 9.58·5-s − 6.51·6-s + 14.1·7-s − 24.5·8-s + 9·9-s + 20.8·10-s + 19.1·11-s + 9.85·12-s + 15.5·13-s + 30.7·14-s − 28.7·15-s − 26.9·16-s + 100.·17-s + 19.5·18-s + 74.3·19-s − 31.4·20-s − 42.4·21-s + 41.5·22-s + 98.9·23-s + 73.5·24-s − 33.2·25-s + 33.7·26-s − 27·27-s − 46.5·28-s + ⋯
L(s)  = 1  + 0.767·2-s − 0.577·3-s − 0.410·4-s + 0.856·5-s − 0.443·6-s + 0.764·7-s − 1.08·8-s + 0.333·9-s + 0.657·10-s + 0.524·11-s + 0.237·12-s + 0.331·13-s + 0.586·14-s − 0.494·15-s − 0.420·16-s + 1.43·17-s + 0.255·18-s + 0.897·19-s − 0.351·20-s − 0.441·21-s + 0.402·22-s + 0.896·23-s + 0.625·24-s − 0.265·25-s + 0.254·26-s − 0.192·27-s − 0.313·28-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

\(L(2)\) \(\approx\) \(2.313855493\)
\(L(\frac12)\) \(\approx\) \(2.313855493\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.17T + 8T^{2} \)
5 \( 1 - 9.58T + 125T^{2} \)
7 \( 1 - 14.1T + 343T^{2} \)
11 \( 1 - 19.1T + 1.33e3T^{2} \)
13 \( 1 - 15.5T + 2.19e3T^{2} \)
17 \( 1 - 100.T + 4.91e3T^{2} \)
19 \( 1 - 74.3T + 6.85e3T^{2} \)
23 \( 1 - 98.9T + 1.21e4T^{2} \)
29 \( 1 - 194.T + 2.43e4T^{2} \)
31 \( 1 - 52.9T + 2.97e4T^{2} \)
37 \( 1 + 212.T + 5.06e4T^{2} \)
41 \( 1 + 395.T + 6.89e4T^{2} \)
43 \( 1 - 305.T + 7.95e4T^{2} \)
47 \( 1 + 630.T + 1.03e5T^{2} \)
53 \( 1 + 109.T + 1.48e5T^{2} \)
61 \( 1 - 240.T + 2.26e5T^{2} \)
67 \( 1 + 100.T + 3.00e5T^{2} \)
71 \( 1 - 263.T + 3.57e5T^{2} \)
73 \( 1 + 296.T + 3.89e5T^{2} \)
79 \( 1 - 626.T + 4.93e5T^{2} \)
83 \( 1 + 7.08T + 5.71e5T^{2} \)
89 \( 1 + 132.T + 7.04e5T^{2} \)
97 \( 1 - 1.47e3T + 9.12e5T^{2} \)
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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

−12.23270159616825908979944028618, −11.57140395204145285764865043073, −10.23014753980857511298500034626, −9.374883977615426624802669257936, −8.151817649124999144923538273422, −6.58366423979733948351723909779, −5.51846518626544078195865130771, −4.85322532971500373509642428321, −3.34833005574944957421137828402, −1.26446824158223135992006581756, 1.26446824158223135992006581756, 3.34833005574944957421137828402, 4.85322532971500373509642428321, 5.51846518626544078195865130771, 6.58366423979733948351723909779, 8.151817649124999144923538273422, 9.374883977615426624802669257936, 10.23014753980857511298500034626, 11.57140395204145285764865043073, 12.23270159616825908979944028618

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