Properties

Label 2-177-1.1-c5-0-41
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.75·2-s + 9·3-s − 9.40·4-s − 8.06·5-s + 42.7·6-s − 28.8·7-s − 196.·8-s + 81·9-s − 38.3·10-s + 556.·11-s − 84.6·12-s − 1.12e3·13-s − 137.·14-s − 72.6·15-s − 634.·16-s + 447.·17-s + 385.·18-s − 1.35e3·19-s + 75.8·20-s − 259.·21-s + 2.64e3·22-s − 4.46e3·23-s − 1.77e3·24-s − 3.05e3·25-s − 5.36e3·26-s + 729·27-s + 271.·28-s + ⋯
L(s)  = 1  + 0.840·2-s + 0.577·3-s − 0.293·4-s − 0.144·5-s + 0.485·6-s − 0.222·7-s − 1.08·8-s + 0.333·9-s − 0.121·10-s + 1.38·11-s − 0.169·12-s − 1.85·13-s − 0.187·14-s − 0.0833·15-s − 0.619·16-s + 0.375·17-s + 0.280·18-s − 0.858·19-s + 0.0424·20-s − 0.128·21-s + 1.16·22-s − 1.76·23-s − 0.627·24-s − 0.979·25-s − 1.55·26-s + 0.192·27-s + 0.0653·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 - 4.75T + 32T^{2} \)
5 \( 1 + 8.06T + 3.12e3T^{2} \)
7 \( 1 + 28.8T + 1.68e4T^{2} \)
11 \( 1 - 556.T + 1.61e5T^{2} \)
13 \( 1 + 1.12e3T + 3.71e5T^{2} \)
17 \( 1 - 447.T + 1.41e6T^{2} \)
19 \( 1 + 1.35e3T + 2.47e6T^{2} \)
23 \( 1 + 4.46e3T + 6.43e6T^{2} \)
29 \( 1 + 2.94e3T + 2.05e7T^{2} \)
31 \( 1 - 1.11e3T + 2.86e7T^{2} \)
37 \( 1 + 5.87e3T + 6.93e7T^{2} \)
41 \( 1 + 638.T + 1.15e8T^{2} \)
43 \( 1 + 9.93e3T + 1.47e8T^{2} \)
47 \( 1 - 2.50e4T + 2.29e8T^{2} \)
53 \( 1 - 1.93e4T + 4.18e8T^{2} \)
61 \( 1 - 7.81e3T + 8.44e8T^{2} \)
67 \( 1 - 3.68e4T + 1.35e9T^{2} \)
71 \( 1 - 2.41e4T + 1.80e9T^{2} \)
73 \( 1 + 7.82e4T + 2.07e9T^{2} \)
79 \( 1 - 4.20e3T + 3.07e9T^{2} \)
83 \( 1 + 3.69e4T + 3.93e9T^{2} \)
89 \( 1 - 6.62e3T + 5.58e9T^{2} \)
97 \( 1 - 8.83e4T + 8.58e9T^{2} \)
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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

−11.90553115923213706387936197344, −10.01736259890681030338762115553, −9.377901907625798017190548097218, −8.257085124332135366338489052186, −6.99620885626459293364338998637, −5.78441360459215798312607384769, −4.41883166826140359608333984377, −3.65704564083840324805145906435, −2.17623096942402966892730741968, 0, 2.17623096942402966892730741968, 3.65704564083840324805145906435, 4.41883166826140359608333984377, 5.78441360459215798312607384769, 6.99620885626459293364338998637, 8.257085124332135366338489052186, 9.377901907625798017190548097218, 10.01736259890681030338762115553, 11.90553115923213706387936197344

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