Properties

Label 2-177-1.1-c9-0-0
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $91.1613$
Root an. cond. $9.54784$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 13.0·2-s − 81·3-s − 342.·4-s − 5.38·5-s − 1.05e3·6-s − 6.26e3·7-s − 1.11e4·8-s + 6.56e3·9-s − 69.9·10-s − 4.36e4·11-s + 2.77e4·12-s − 6.91e4·13-s − 8.14e4·14-s + 436.·15-s + 3.11e4·16-s − 3.25e5·17-s + 8.52e4·18-s − 6.28e5·19-s + 1.84e3·20-s + 5.07e5·21-s − 5.66e5·22-s − 4.08e5·23-s + 9.00e5·24-s − 1.95e6·25-s − 8.98e5·26-s − 5.31e5·27-s + 2.14e6·28-s + ⋯
L(s)  = 1  + 0.574·2-s − 0.577·3-s − 0.669·4-s − 0.00385·5-s − 0.331·6-s − 0.985·7-s − 0.959·8-s + 0.333·9-s − 0.00221·10-s − 0.898·11-s + 0.386·12-s − 0.671·13-s − 0.566·14-s + 0.00222·15-s + 0.118·16-s − 0.945·17-s + 0.191·18-s − 1.10·19-s + 0.00258·20-s + 0.569·21-s − 0.516·22-s − 0.304·23-s + 0.553·24-s − 0.999·25-s − 0.385·26-s − 0.192·27-s + 0.660·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(91.1613\)
Root analytic conductor: \(9.54784\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.01551290879\)
\(L(\frac12)\) \(\approx\) \(0.01551290879\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 81T \)
59 \( 1 - 1.21e7T \)
good2 \( 1 - 13.0T + 512T^{2} \)
5 \( 1 + 5.38T + 1.95e6T^{2} \)
7 \( 1 + 6.26e3T + 4.03e7T^{2} \)
11 \( 1 + 4.36e4T + 2.35e9T^{2} \)
13 \( 1 + 6.91e4T + 1.06e10T^{2} \)
17 \( 1 + 3.25e5T + 1.18e11T^{2} \)
19 \( 1 + 6.28e5T + 3.22e11T^{2} \)
23 \( 1 + 4.08e5T + 1.80e12T^{2} \)
29 \( 1 + 7.39e6T + 1.45e13T^{2} \)
31 \( 1 - 6.85e5T + 2.64e13T^{2} \)
37 \( 1 - 4.07e6T + 1.29e14T^{2} \)
41 \( 1 + 2.00e6T + 3.27e14T^{2} \)
43 \( 1 - 5.29e6T + 5.02e14T^{2} \)
47 \( 1 + 4.79e7T + 1.11e15T^{2} \)
53 \( 1 - 2.75e7T + 3.29e15T^{2} \)
61 \( 1 + 1.32e8T + 1.16e16T^{2} \)
67 \( 1 + 8.56e7T + 2.72e16T^{2} \)
71 \( 1 - 1.57e8T + 4.58e16T^{2} \)
73 \( 1 - 3.62e8T + 5.88e16T^{2} \)
79 \( 1 - 1.14e7T + 1.19e17T^{2} \)
83 \( 1 + 4.93e7T + 1.86e17T^{2} \)
89 \( 1 - 8.84e8T + 3.50e17T^{2} \)
97 \( 1 + 7.61e8T + 7.60e17T^{2} \)
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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

−11.08187483002978631751960095091, −9.971122796864254818503355280958, −9.221417924328501794120117486953, −7.894164810721956530830122131257, −6.54654579803220586211712714093, −5.66963008600234600939654774186, −4.64105490836701662928814818262, −3.62244763503560949269388741171, −2.25557841807134978345707101436, −0.05402096607047457878237354645, 0.05402096607047457878237354645, 2.25557841807134978345707101436, 3.62244763503560949269388741171, 4.64105490836701662928814818262, 5.66963008600234600939654774186, 6.54654579803220586211712714093, 7.894164810721956530830122131257, 9.221417924328501794120117486953, 9.971122796864254818503355280958, 11.08187483002978631751960095091

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