Properties

Label 2-177-1.1-c3-0-4
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  − 3.77·2-s + 3·3-s + 6.27·4-s − 5.74·5-s − 11.3·6-s − 24.4·7-s + 6.51·8-s + 9·9-s + 21.6·10-s + 9.31·11-s + 18.8·12-s + 17.0·13-s + 92.2·14-s − 17.2·15-s − 74.8·16-s + 19.6·17-s − 34.0·18-s − 15.1·19-s − 36.0·20-s − 73.2·21-s − 35.2·22-s + 108.·23-s + 19.5·24-s − 92.0·25-s − 64.5·26-s + 27·27-s − 153.·28-s + ⋯
L(s)  = 1  − 1.33·2-s + 0.577·3-s + 0.784·4-s − 0.513·5-s − 0.771·6-s − 1.31·7-s + 0.287·8-s + 0.333·9-s + 0.686·10-s + 0.255·11-s + 0.452·12-s + 0.364·13-s + 1.76·14-s − 0.296·15-s − 1.16·16-s + 0.280·17-s − 0.445·18-s − 0.182·19-s − 0.403·20-s − 0.761·21-s − 0.341·22-s + 0.983·23-s + 0.166·24-s − 0.736·25-s − 0.486·26-s + 0.192·27-s − 1.03·28-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

\(L(2)\) \(\approx\) \(0.7770742793\)
\(L(\frac12)\) \(\approx\) \(0.7770742793\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.77T + 8T^{2} \)
5 \( 1 + 5.74T + 125T^{2} \)
7 \( 1 + 24.4T + 343T^{2} \)
11 \( 1 - 9.31T + 1.33e3T^{2} \)
13 \( 1 - 17.0T + 2.19e3T^{2} \)
17 \( 1 - 19.6T + 4.91e3T^{2} \)
19 \( 1 + 15.1T + 6.85e3T^{2} \)
23 \( 1 - 108.T + 1.21e4T^{2} \)
29 \( 1 - 237.T + 2.43e4T^{2} \)
31 \( 1 - 184.T + 2.97e4T^{2} \)
37 \( 1 - 155.T + 5.06e4T^{2} \)
41 \( 1 - 261.T + 6.89e4T^{2} \)
43 \( 1 - 127.T + 7.95e4T^{2} \)
47 \( 1 - 555.T + 1.03e5T^{2} \)
53 \( 1 + 220.T + 1.48e5T^{2} \)
61 \( 1 + 436.T + 2.26e5T^{2} \)
67 \( 1 - 924.T + 3.00e5T^{2} \)
71 \( 1 + 937.T + 3.57e5T^{2} \)
73 \( 1 - 13.0T + 3.89e5T^{2} \)
79 \( 1 - 140.T + 4.93e5T^{2} \)
83 \( 1 + 26.4T + 5.71e5T^{2} \)
89 \( 1 + 136.T + 7.04e5T^{2} \)
97 \( 1 + 360.T + 9.12e5T^{2} \)
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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

−12.13758648088718899430756116849, −10.86752536851610274705329744082, −9.923579445399687977090867137471, −9.210379500379979198709286873142, −8.335019884976794213915210914548, −7.36312692437001877978665851264, −6.34293714216765274886166877907, −4.22159345571315775540981695953, −2.81465768932797624550635757335, −0.828348935450798559131087068459, 0.828348935450798559131087068459, 2.81465768932797624550635757335, 4.22159345571315775540981695953, 6.34293714216765274886166877907, 7.36312692437001877978665851264, 8.335019884976794213915210914548, 9.210379500379979198709286873142, 9.923579445399687977090867137471, 10.86752536851610274705329744082, 12.13758648088718899430756116849

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