Properties

Label 2-177-1.1-c9-0-75
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.9·2-s + 81·3-s − 256.·4-s + 61.0·5-s + 1.29e3·6-s + 4.75e3·7-s − 1.22e4·8-s + 6.56e3·9-s + 976.·10-s + 5.60e3·11-s − 2.07e4·12-s − 7.34e4·13-s + 7.60e4·14-s + 4.94e3·15-s − 6.49e4·16-s + 4.30e5·17-s + 1.04e5·18-s − 7.18e5·19-s − 1.56e4·20-s + 3.85e5·21-s + 8.95e4·22-s + 3.94e3·23-s − 9.94e5·24-s − 1.94e6·25-s − 1.17e6·26-s + 5.31e5·27-s − 1.22e6·28-s + ⋯
L(s)  = 1  + 0.706·2-s + 0.577·3-s − 0.501·4-s + 0.0437·5-s + 0.407·6-s + 0.748·7-s − 1.06·8-s + 0.333·9-s + 0.0308·10-s + 0.115·11-s − 0.289·12-s − 0.713·13-s + 0.528·14-s + 0.0252·15-s − 0.247·16-s + 1.25·17-s + 0.235·18-s − 1.26·19-s − 0.0219·20-s + 0.432·21-s + 0.0815·22-s + 0.00293·23-s − 0.612·24-s − 0.998·25-s − 0.504·26-s + 0.192·27-s − 0.375·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 - 15.9T + 512T^{2} \)
5 \( 1 - 61.0T + 1.95e6T^{2} \)
7 \( 1 - 4.75e3T + 4.03e7T^{2} \)
11 \( 1 - 5.60e3T + 2.35e9T^{2} \)
13 \( 1 + 7.34e4T + 1.06e10T^{2} \)
17 \( 1 - 4.30e5T + 1.18e11T^{2} \)
19 \( 1 + 7.18e5T + 3.22e11T^{2} \)
23 \( 1 - 3.94e3T + 1.80e12T^{2} \)
29 \( 1 - 6.27e5T + 1.45e13T^{2} \)
31 \( 1 - 1.02e6T + 2.64e13T^{2} \)
37 \( 1 + 1.31e7T + 1.29e14T^{2} \)
41 \( 1 + 7.00e6T + 3.27e14T^{2} \)
43 \( 1 - 3.11e7T + 5.02e14T^{2} \)
47 \( 1 + 7.14e6T + 1.11e15T^{2} \)
53 \( 1 + 1.00e8T + 3.29e15T^{2} \)
61 \( 1 + 4.94e7T + 1.16e16T^{2} \)
67 \( 1 + 2.76e8T + 2.72e16T^{2} \)
71 \( 1 + 2.12e8T + 4.58e16T^{2} \)
73 \( 1 + 1.63e8T + 5.88e16T^{2} \)
79 \( 1 + 5.43e8T + 1.19e17T^{2} \)
83 \( 1 - 5.64e8T + 1.86e17T^{2} \)
89 \( 1 + 3.74e8T + 3.50e17T^{2} \)
97 \( 1 - 1.37e9T + 7.60e17T^{2} \)
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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

−10.42995156718955564209807906832, −9.442618692062876935740685275326, −8.457817418582118659456516983570, −7.57832494297111148478085462486, −6.07292271037448532873155609639, −4.95856862870034185793564683247, −4.09013506190733766332922435443, −2.94534865491226763677713229812, −1.61629084179723735324273536812, 0, 1.61629084179723735324273536812, 2.94534865491226763677713229812, 4.09013506190733766332922435443, 4.95856862870034185793564683247, 6.07292271037448532873155609639, 7.57832494297111148478085462486, 8.457817418582118659456516983570, 9.442618692062876935740685275326, 10.42995156718955564209807906832

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