Properties

Label 2-177-1.1-c5-0-9
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.47·2-s − 9·3-s + 57.7·4-s − 42.5·5-s + 85.2·6-s + 243.·7-s − 243.·8-s + 81·9-s + 402.·10-s + 382.·11-s − 519.·12-s + 434.·13-s − 2.30e3·14-s + 382.·15-s + 461.·16-s + 67.7·17-s − 767.·18-s + 2.84e3·19-s − 2.45e3·20-s − 2.18e3·21-s − 3.62e3·22-s − 1.47e3·23-s + 2.19e3·24-s − 1.31e3·25-s − 4.11e3·26-s − 729·27-s + 1.40e4·28-s + ⋯
L(s)  = 1  − 1.67·2-s − 0.577·3-s + 1.80·4-s − 0.760·5-s + 0.966·6-s + 1.87·7-s − 1.34·8-s + 0.333·9-s + 1.27·10-s + 0.953·11-s − 1.04·12-s + 0.713·13-s − 3.13·14-s + 0.439·15-s + 0.450·16-s + 0.0568·17-s − 0.558·18-s + 1.80·19-s − 1.37·20-s − 1.08·21-s − 1.59·22-s − 0.582·23-s + 0.777·24-s − 0.421·25-s − 1.19·26-s − 0.192·27-s + 3.38·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8594190800\)
\(L(\frac12)\) \(\approx\) \(0.8594190800\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 9T \)
59 \( 1 - 3.48e3T \)
good2 \( 1 + 9.47T + 32T^{2} \)
5 \( 1 + 42.5T + 3.12e3T^{2} \)
7 \( 1 - 243.T + 1.68e4T^{2} \)
11 \( 1 - 382.T + 1.61e5T^{2} \)
13 \( 1 - 434.T + 3.71e5T^{2} \)
17 \( 1 - 67.7T + 1.41e6T^{2} \)
19 \( 1 - 2.84e3T + 2.47e6T^{2} \)
23 \( 1 + 1.47e3T + 6.43e6T^{2} \)
29 \( 1 + 413.T + 2.05e7T^{2} \)
31 \( 1 + 7.56e3T + 2.86e7T^{2} \)
37 \( 1 - 2.85e3T + 6.93e7T^{2} \)
41 \( 1 - 4.30e3T + 1.15e8T^{2} \)
43 \( 1 - 991.T + 1.47e8T^{2} \)
47 \( 1 + 6.81e3T + 2.29e8T^{2} \)
53 \( 1 - 2.79e4T + 4.18e8T^{2} \)
61 \( 1 - 3.29e4T + 8.44e8T^{2} \)
67 \( 1 + 1.44e4T + 1.35e9T^{2} \)
71 \( 1 - 3.15e4T + 1.80e9T^{2} \)
73 \( 1 - 2.12e4T + 2.07e9T^{2} \)
79 \( 1 + 5.10e4T + 3.07e9T^{2} \)
83 \( 1 + 1.78e4T + 3.93e9T^{2} \)
89 \( 1 + 3.88e4T + 5.58e9T^{2} \)
97 \( 1 + 8.32e4T + 8.58e9T^{2} \)
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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

−11.44639266773029520775411820421, −10.97606489544409441703886756590, −9.707879430516852599188142668550, −8.644565155991449327133934784241, −7.83005574584808214989960666515, −7.14013905472905383690133860813, −5.54143750360668208053642586861, −4.04365341264791243718414907918, −1.70137145366776843658678035656, −0.850422553592304083520457174587, 0.850422553592304083520457174587, 1.70137145366776843658678035656, 4.04365341264791243718414907918, 5.54143750360668208053642586861, 7.14013905472905383690133860813, 7.83005574584808214989960666515, 8.644565155991449327133934784241, 9.707879430516852599188142668550, 10.97606489544409441703886756590, 11.44639266773029520775411820421

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