Properties

Label 2-177-1.1-c9-0-19
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  + 22.7·2-s + 81·3-s + 5.71·4-s − 1.39e3·5-s + 1.84e3·6-s − 3.35e3·7-s − 1.15e4·8-s + 6.56e3·9-s − 3.18e4·10-s + 3.56e4·11-s + 462.·12-s − 1.47e5·13-s − 7.63e4·14-s − 1.13e5·15-s − 2.65e5·16-s + 6.51e4·17-s + 1.49e5·18-s + 6.99e5·19-s − 7.98e3·20-s − 2.71e5·21-s + 8.10e5·22-s + 9.96e5·23-s − 9.33e5·24-s + 1.17e3·25-s − 3.36e6·26-s + 5.31e5·27-s − 1.91e4·28-s + ⋯
L(s)  = 1  + 1.00·2-s + 0.577·3-s + 0.0111·4-s − 1.00·5-s + 0.580·6-s − 0.527·7-s − 0.994·8-s + 0.333·9-s − 1.00·10-s + 0.733·11-s + 0.00643·12-s − 1.43·13-s − 0.530·14-s − 0.577·15-s − 1.01·16-s + 0.189·17-s + 0.335·18-s + 1.23·19-s − 0.0111·20-s − 0.304·21-s + 0.737·22-s + 0.742·23-s − 0.574·24-s + 0.000603·25-s − 1.44·26-s + 0.192·27-s − 0.00588·28-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

\(L(5)\) \(\approx\) \(2.560001714\)
\(L(\frac12)\) \(\approx\) \(2.560001714\)
\(L(\frac{11}{2})\) 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 \( 1 - 22.7T + 512T^{2} \)
5 \( 1 + 1.39e3T + 1.95e6T^{2} \)
7 \( 1 + 3.35e3T + 4.03e7T^{2} \)
11 \( 1 - 3.56e4T + 2.35e9T^{2} \)
13 \( 1 + 1.47e5T + 1.06e10T^{2} \)
17 \( 1 - 6.51e4T + 1.18e11T^{2} \)
19 \( 1 - 6.99e5T + 3.22e11T^{2} \)
23 \( 1 - 9.96e5T + 1.80e12T^{2} \)
29 \( 1 + 3.88e6T + 1.45e13T^{2} \)
31 \( 1 - 2.43e6T + 2.64e13T^{2} \)
37 \( 1 - 1.41e7T + 1.29e14T^{2} \)
41 \( 1 + 1.02e7T + 3.27e14T^{2} \)
43 \( 1 - 2.85e7T + 5.02e14T^{2} \)
47 \( 1 - 2.87e7T + 1.11e15T^{2} \)
53 \( 1 - 7.51e7T + 3.29e15T^{2} \)
61 \( 1 - 1.55e8T + 1.16e16T^{2} \)
67 \( 1 + 1.39e8T + 2.72e16T^{2} \)
71 \( 1 - 1.41e8T + 4.58e16T^{2} \)
73 \( 1 - 2.22e8T + 5.88e16T^{2} \)
79 \( 1 + 7.34e7T + 1.19e17T^{2} \)
83 \( 1 + 1.06e8T + 1.86e17T^{2} \)
89 \( 1 + 1.11e7T + 3.50e17T^{2} \)
97 \( 1 + 1.97e8T + 7.60e17T^{2} \)
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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

−11.48508863188760930774824147190, −9.771529444101089863856444524498, −9.121035659088070452976706435644, −7.78414030147428607502583371139, −6.91704730441407557577765068186, −5.47856008268203536363432191437, −4.35428518172429676911507559893, −3.55832479712604342348656766642, −2.62942764462855449818874889803, −0.64504251349180628997191570251, 0.64504251349180628997191570251, 2.62942764462855449818874889803, 3.55832479712604342348656766642, 4.35428518172429676911507559893, 5.47856008268203536363432191437, 6.91704730441407557577765068186, 7.78414030147428607502583371139, 9.121035659088070452976706435644, 9.771529444101089863856444524498, 11.48508863188760930774824147190

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