Properties

Label 2-177-1.1-c5-0-12
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  + 0.615·2-s + 9·3-s − 31.6·4-s + 61.9·5-s + 5.53·6-s − 209.·7-s − 39.1·8-s + 81·9-s + 38.1·10-s + 1.45·11-s − 284.·12-s + 351.·13-s − 128.·14-s + 557.·15-s + 987.·16-s + 689.·17-s + 49.8·18-s + 2.73e3·19-s − 1.95e3·20-s − 1.88e3·21-s + 0.895·22-s − 1.31e3·23-s − 352.·24-s + 711.·25-s + 216.·26-s + 729·27-s + 6.62e3·28-s + ⋯
L(s)  = 1  + 0.108·2-s + 0.577·3-s − 0.988·4-s + 1.10·5-s + 0.0628·6-s − 1.61·7-s − 0.216·8-s + 0.333·9-s + 0.120·10-s + 0.00362·11-s − 0.570·12-s + 0.576·13-s − 0.175·14-s + 0.639·15-s + 0.964·16-s + 0.578·17-s + 0.0362·18-s + 1.73·19-s − 1.09·20-s − 0.933·21-s + 0.000394·22-s − 0.516·23-s − 0.124·24-s + 0.227·25-s + 0.0627·26-s + 0.192·27-s + 1.59·28-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

\(L(3)\) \(\approx\) \(2.161211866\)
\(L(\frac12)\) \(\approx\) \(2.161211866\)
\(L(\frac{7}{2})\) 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 \( 1 - 0.615T + 32T^{2} \)
5 \( 1 - 61.9T + 3.12e3T^{2} \)
7 \( 1 + 209.T + 1.68e4T^{2} \)
11 \( 1 - 1.45T + 1.61e5T^{2} \)
13 \( 1 - 351.T + 3.71e5T^{2} \)
17 \( 1 - 689.T + 1.41e6T^{2} \)
19 \( 1 - 2.73e3T + 2.47e6T^{2} \)
23 \( 1 + 1.31e3T + 6.43e6T^{2} \)
29 \( 1 - 508.T + 2.05e7T^{2} \)
31 \( 1 - 4.21e3T + 2.86e7T^{2} \)
37 \( 1 - 4.02e3T + 6.93e7T^{2} \)
41 \( 1 - 7.33e3T + 1.15e8T^{2} \)
43 \( 1 - 5.25e3T + 1.47e8T^{2} \)
47 \( 1 - 2.53e4T + 2.29e8T^{2} \)
53 \( 1 + 1.65e4T + 4.18e8T^{2} \)
61 \( 1 + 4.26e3T + 8.44e8T^{2} \)
67 \( 1 - 1.33e4T + 1.35e9T^{2} \)
71 \( 1 - 5.41e4T + 1.80e9T^{2} \)
73 \( 1 + 3.25e4T + 2.07e9T^{2} \)
79 \( 1 - 5.95e4T + 3.07e9T^{2} \)
83 \( 1 - 9.40e4T + 3.93e9T^{2} \)
89 \( 1 + 1.36e5T + 5.58e9T^{2} \)
97 \( 1 - 1.35e5T + 8.58e9T^{2} \)
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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

−12.19652148076792358221733907555, −10.29546505742283704025098608503, −9.571013754296111425047454830042, −9.194166131776140881912040999251, −7.79012814704782905740070857240, −6.31471176896636702149528044085, −5.47265288674357907631909711682, −3.82506940322294436090286179966, −2.83310834183008786614299267827, −0.930176398529957473961550683274, 0.930176398529957473961550683274, 2.83310834183008786614299267827, 3.82506940322294436090286179966, 5.47265288674357907631909711682, 6.31471176896636702149528044085, 7.79012814704782905740070857240, 9.194166131776140881912040999251, 9.571013754296111425047454830042, 10.29546505742283704025098608503, 12.19652148076792358221733907555

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