Properties

Degree $2$
Conductor $177$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.11·2-s − 3-s + 2.47·4-s − 0.357·5-s − 2.11·6-s + 5.11·7-s + 1.00·8-s + 9-s − 0.756·10-s − 4.58·11-s − 2.47·12-s − 2.58·13-s + 10.8·14-s + 0.357·15-s − 2.83·16-s + 2.18·17-s + 2.11·18-s + 0.527·19-s − 0.885·20-s − 5.11·21-s − 9.70·22-s − 5.70·23-s − 1.00·24-s − 4.87·25-s − 5.47·26-s − 27-s + 12.6·28-s + ⋯
L(s)  = 1  + 1.49·2-s − 0.577·3-s + 1.23·4-s − 0.160·5-s − 0.863·6-s + 1.93·7-s + 0.353·8-s + 0.333·9-s − 0.239·10-s − 1.38·11-s − 0.713·12-s − 0.717·13-s + 2.89·14-s + 0.0924·15-s − 0.707·16-s + 0.530·17-s + 0.498·18-s + 0.120·19-s − 0.197·20-s − 1.11·21-s − 2.06·22-s − 1.18·23-s − 0.204·24-s − 0.974·25-s − 1.07·26-s − 0.192·27-s + 2.39·28-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: $1$
Motivic weight: \(1\)
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09612\)
\(L(\frac12)\) \(\approx\) \(2.09612\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.11T + 2T^{2} \)
5 \( 1 + 0.357T + 5T^{2} \)
7 \( 1 - 5.11T + 7T^{2} \)
11 \( 1 + 4.58T + 11T^{2} \)
13 \( 1 + 2.58T + 13T^{2} \)
17 \( 1 - 2.18T + 17T^{2} \)
19 \( 1 - 0.527T + 19T^{2} \)
23 \( 1 + 5.70T + 23T^{2} \)
29 \( 1 - 2.06T + 29T^{2} \)
31 \( 1 - 5.83T + 31T^{2} \)
37 \( 1 + 7.70T + 37T^{2} \)
41 \( 1 - 2.39T + 41T^{2} \)
43 \( 1 - 7.15T + 43T^{2} \)
47 \( 1 - 8.77T + 47T^{2} \)
53 \( 1 + 8.10T + 53T^{2} \)
61 \( 1 - 9.00T + 61T^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 - 4.12T + 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 + 3.87T + 79T^{2} \)
83 \( 1 + 0.737T + 83T^{2} \)
89 \( 1 + 8.54T + 89T^{2} \)
97 \( 1 + 4.10T + 97T^{2} \)
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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

−12.55618185611442218092884749135, −11.91684371954860367917072229032, −11.15985552852358300343642681868, −10.16212449178790756649317697097, −8.216604281349738454771958114708, −7.39228872938730447510477628252, −5.72611373648574677162227333751, −5.08367229387006211553320505309, −4.21342416017582223945039289972, −2.31017527179416978223257425965, 2.31017527179416978223257425965, 4.21342416017582223945039289972, 5.08367229387006211553320505309, 5.72611373648574677162227333751, 7.39228872938730447510477628252, 8.216604281349738454771958114708, 10.16212449178790756649317697097, 11.15985552852358300343642681868, 11.91684371954860367917072229032, 12.55618185611442218092884749135

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