Properties

Label 2-177-1.1-c13-0-121
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $189.798$
Root an. cond. $13.7767$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 124.·2-s + 729·3-s + 7.27e3·4-s + 5.02e4·5-s + 9.06e4·6-s + 1.88e4·7-s − 1.14e5·8-s + 5.31e5·9-s + 6.24e6·10-s − 3.50e6·11-s + 5.30e6·12-s − 3.23e7·13-s + 2.34e6·14-s + 3.66e7·15-s − 7.37e7·16-s − 5.75e7·17-s + 6.60e7·18-s + 5.27e7·19-s + 3.65e8·20-s + 1.37e7·21-s − 4.36e8·22-s + 1.20e8·23-s − 8.34e7·24-s + 1.30e9·25-s − 4.02e9·26-s + 3.87e8·27-s + 1.37e8·28-s + ⋯
L(s)  = 1  + 1.37·2-s + 0.577·3-s + 0.887·4-s + 1.43·5-s + 0.793·6-s + 0.0606·7-s − 0.154·8-s + 0.333·9-s + 1.97·10-s − 0.597·11-s + 0.512·12-s − 1.85·13-s + 0.0833·14-s + 0.830·15-s − 1.09·16-s − 0.577·17-s + 0.457·18-s + 0.257·19-s + 1.27·20-s + 0.0350·21-s − 0.820·22-s + 0.169·23-s − 0.0890·24-s + 1.06·25-s − 2.55·26-s + 0.192·27-s + 0.0538·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(189.798\)
Root analytic conductor: \(13.7767\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 729T \)
59 \( 1 - 4.21e10T \)
good2 \( 1 - 124.T + 8.19e3T^{2} \)
5 \( 1 - 5.02e4T + 1.22e9T^{2} \)
7 \( 1 - 1.88e4T + 9.68e10T^{2} \)
11 \( 1 + 3.50e6T + 3.45e13T^{2} \)
13 \( 1 + 3.23e7T + 3.02e14T^{2} \)
17 \( 1 + 5.75e7T + 9.90e15T^{2} \)
19 \( 1 - 5.27e7T + 4.20e16T^{2} \)
23 \( 1 - 1.20e8T + 5.04e17T^{2} \)
29 \( 1 + 5.16e9T + 1.02e19T^{2} \)
31 \( 1 + 3.59e9T + 2.44e19T^{2} \)
37 \( 1 + 2.50e10T + 2.43e20T^{2} \)
41 \( 1 - 1.78e10T + 9.25e20T^{2} \)
43 \( 1 - 1.95e10T + 1.71e21T^{2} \)
47 \( 1 + 2.54e10T + 5.46e21T^{2} \)
53 \( 1 + 1.37e11T + 2.60e22T^{2} \)
61 \( 1 - 1.44e11T + 1.61e23T^{2} \)
67 \( 1 - 5.79e11T + 5.48e23T^{2} \)
71 \( 1 - 4.08e11T + 1.16e24T^{2} \)
73 \( 1 - 1.35e11T + 1.67e24T^{2} \)
79 \( 1 - 4.13e12T + 4.66e24T^{2} \)
83 \( 1 - 2.23e12T + 8.87e24T^{2} \)
89 \( 1 - 7.74e12T + 2.19e25T^{2} \)
97 \( 1 - 8.32e12T + 6.73e25T^{2} \)
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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

−9.713339760041861022035187585564, −9.152301426106370368056821237094, −7.54803738753279343216733219576, −6.53417112147435215426633007169, −5.36015069377804462751580565904, −4.91885898706426733912836092869, −3.52336161292319300809753421333, −2.41728715780941117761961036306, −1.97126813911182542834397864611, 0, 1.97126813911182542834397864611, 2.41728715780941117761961036306, 3.52336161292319300809753421333, 4.91885898706426733912836092869, 5.36015069377804462751580565904, 6.53417112147435215426633007169, 7.54803738753279343216733219576, 9.152301426106370368056821237094, 9.713339760041861022035187585564

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