Properties

Label 2-177-1.1-c1-0-2
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $1.41335$
Root an. cond. $1.18884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.618·2-s + 3-s − 1.61·4-s + 5-s − 0.618·6-s + 1.61·7-s + 2.23·8-s + 9-s − 0.618·10-s + 4.23·11-s − 1.61·12-s − 13-s − 1.00·14-s + 15-s + 1.85·16-s + 3.85·17-s − 0.618·18-s − 5.85·19-s − 1.61·20-s + 1.61·21-s − 2.61·22-s − 1.85·23-s + 2.23·24-s − 4·25-s + 0.618·26-s + 27-s − 2.61·28-s + ⋯
L(s)  = 1  − 0.437·2-s + 0.577·3-s − 0.809·4-s + 0.447·5-s − 0.252·6-s + 0.611·7-s + 0.790·8-s + 0.333·9-s − 0.195·10-s + 1.27·11-s − 0.467·12-s − 0.277·13-s − 0.267·14-s + 0.258·15-s + 0.463·16-s + 0.934·17-s − 0.145·18-s − 1.34·19-s − 0.361·20-s + 0.353·21-s − 0.558·22-s − 0.386·23-s + 0.456·24-s − 0.800·25-s + 0.121·26-s + 0.192·27-s − 0.494·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.106657520\)
\(L(\frac12)\) \(\approx\) \(1.106657520\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
59 \( 1 + T \)
good2 \( 1 + 0.618T + 2T^{2} \)
5 \( 1 - T + 5T^{2} \)
7 \( 1 - 1.61T + 7T^{2} \)
11 \( 1 - 4.23T + 11T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 - 3.85T + 17T^{2} \)
19 \( 1 + 5.85T + 19T^{2} \)
23 \( 1 + 1.85T + 23T^{2} \)
29 \( 1 - 3.61T + 29T^{2} \)
31 \( 1 - 2.85T + 31T^{2} \)
37 \( 1 + 3.38T + 37T^{2} \)
41 \( 1 + 6.09T + 41T^{2} \)
43 \( 1 + 9.94T + 43T^{2} \)
47 \( 1 - 4.38T + 47T^{2} \)
53 \( 1 + 9.94T + 53T^{2} \)
61 \( 1 - 2.85T + 61T^{2} \)
67 \( 1 - 11.9T + 67T^{2} \)
71 \( 1 + 4.70T + 71T^{2} \)
73 \( 1 - 0.381T + 73T^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 - 3.14T + 83T^{2} \)
89 \( 1 + 8.09T + 89T^{2} \)
97 \( 1 + 8.70T + 97T^{2} \)
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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

−12.78718315856322729734775489663, −11.75614757311944988792946563884, −10.32256216792915208114197744953, −9.622032481593677660586668674060, −8.641609712175748105035069728924, −7.939745056671385798664398401695, −6.48237622552898002112124554721, −4.93190557476243560191271324132, −3.78102691417589422800003718291, −1.67016810813577794666411149480, 1.67016810813577794666411149480, 3.78102691417589422800003718291, 4.93190557476243560191271324132, 6.48237622552898002112124554721, 7.939745056671385798664398401695, 8.641609712175748105035069728924, 9.622032481593677660586668674060, 10.32256216792915208114197744953, 11.75614757311944988792946563884, 12.78718315856322729734775489663

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