Properties

Label 2-177-1.1-c9-0-16
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  − 38.2·2-s − 81·3-s + 948.·4-s − 2.36e3·5-s + 3.09e3·6-s + 4.56e3·7-s − 1.66e4·8-s + 6.56e3·9-s + 9.03e4·10-s − 4.82e4·11-s − 7.68e4·12-s + 1.91e5·13-s − 1.74e5·14-s + 1.91e5·15-s + 1.51e5·16-s + 3.38e5·17-s − 2.50e5·18-s + 5.24e5·19-s − 2.24e6·20-s − 3.69e5·21-s + 1.84e6·22-s − 1.14e6·23-s + 1.35e6·24-s + 3.63e6·25-s − 7.30e6·26-s − 5.31e5·27-s + 4.32e6·28-s + ⋯
L(s)  = 1  − 1.68·2-s − 0.577·3-s + 1.85·4-s − 1.69·5-s + 0.975·6-s + 0.718·7-s − 1.43·8-s + 0.333·9-s + 2.85·10-s − 0.993·11-s − 1.06·12-s + 1.85·13-s − 1.21·14-s + 0.976·15-s + 0.578·16-s + 0.981·17-s − 0.562·18-s + 0.923·19-s − 3.13·20-s − 0.414·21-s + 1.67·22-s − 0.853·23-s + 0.830·24-s + 1.86·25-s − 3.13·26-s − 0.192·27-s + 1.33·28-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

\(L(5)\) \(\approx\) \(0.5849982144\)
\(L(\frac12)\) \(\approx\) \(0.5849982144\)
\(L(\frac{11}{2})\) 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 \( 1 + 38.2T + 512T^{2} \)
5 \( 1 + 2.36e3T + 1.95e6T^{2} \)
7 \( 1 - 4.56e3T + 4.03e7T^{2} \)
11 \( 1 + 4.82e4T + 2.35e9T^{2} \)
13 \( 1 - 1.91e5T + 1.06e10T^{2} \)
17 \( 1 - 3.38e5T + 1.18e11T^{2} \)
19 \( 1 - 5.24e5T + 3.22e11T^{2} \)
23 \( 1 + 1.14e6T + 1.80e12T^{2} \)
29 \( 1 - 5.85e6T + 1.45e13T^{2} \)
31 \( 1 - 5.88e6T + 2.64e13T^{2} \)
37 \( 1 + 1.22e7T + 1.29e14T^{2} \)
41 \( 1 - 3.14e7T + 3.27e14T^{2} \)
43 \( 1 - 1.58e7T + 5.02e14T^{2} \)
47 \( 1 + 1.12e7T + 1.11e15T^{2} \)
53 \( 1 + 4.33e7T + 3.29e15T^{2} \)
61 \( 1 + 1.55e8T + 1.16e16T^{2} \)
67 \( 1 + 1.33e8T + 2.72e16T^{2} \)
71 \( 1 + 1.02e8T + 4.58e16T^{2} \)
73 \( 1 + 2.90e8T + 5.88e16T^{2} \)
79 \( 1 - 3.11e8T + 1.19e17T^{2} \)
83 \( 1 - 4.30e8T + 1.86e17T^{2} \)
89 \( 1 + 3.66e8T + 3.50e17T^{2} \)
97 \( 1 + 7.50e8T + 7.60e17T^{2} \)
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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

−10.91829734733904979937644358187, −10.20706889978774782312228003083, −8.690964235133468992763103718992, −7.965475275911062953931117875941, −7.54453085507495987715186168367, −6.13783790170146902614330306986, −4.56795141801899934493984337286, −3.16129821677946523038976385323, −1.28269797371452426955810623926, −0.58851901178897894355165023411, 0.58851901178897894355165023411, 1.28269797371452426955810623926, 3.16129821677946523038976385323, 4.56795141801899934493984337286, 6.13783790170146902614330306986, 7.54453085507495987715186168367, 7.965475275911062953931117875941, 8.690964235133468992763103718992, 10.20706889978774782312228003083, 10.91829734733904979937644358187

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